id sid tid token lemma pos american_scientific_journal-610 1 1 american american PROPN american_scientific_journal-610 1 2 scientific scientific ADJ american_scientific_journal-610 1 3 research research NOUN american_scientific_journal-610 1 4 journal journal NOUN american_scientific_journal-610 1 5 for for ADP american_scientific_journal-610 1 6 engineering engineering NOUN american_scientific_journal-610 1 7 , , PUNCT american_scientific_journal-610 1 8 technology technology NOUN american_scientific_journal-610 1 9 , , PUNCT american_scientific_journal-610 1 10 and and CCONJ american_scientific_journal-610 1 11 sciences science NOUN american_scientific_journal-610 1 12 ( ( PUNCT american_scientific_journal-610 1 13 asrjets asrjet NOUN american_scientific_journal-610 1 14 ) ) PUNCT american_scientific_journal-610 1 15 issn issn PROPN american_scientific_journal-610 1 16 ( ( PUNCT american_scientific_journal-610 1 17 print print NOUN american_scientific_journal-610 1 18 ) ) PUNCT american_scientific_journal-610 1 19 2313 2313 NUM american_scientific_journal-610 1 20 - - SYM american_scientific_journal-610 1 21 4410 4410 NUM american_scientific_journal-610 1 22 , , PUNCT american_scientific_journal-610 1 23 issn issn PROPN american_scientific_journal-610 1 24 ( ( PUNCT american_scientific_journal-610 1 25 online online ADJ american_scientific_journal-610 1 26 ) ) PUNCT american_scientific_journal-610 1 27 2313 2313 NUM american_scientific_journal-610 1 28 - - SYM american_scientific_journal-610 1 29 4402 4402 NUM american_scientific_journal-610 1 30 Β© Β© PROPN american_scientific_journal-610 1 31 global global ADJ american_scientific_journal-610 1 32 society society NOUN american_scientific_journal-610 1 33 of of ADP american_scientific_journal-610 1 34 scientific scientific ADJ american_scientific_journal-610 1 35 research research NOUN american_scientific_journal-610 1 36 and and CCONJ american_scientific_journal-610 1 37 researchers researcher NOUN american_scientific_journal-610 1 38 http://asrjetsjournal.org/ http://asrjetsjournal.org/ NOUN american_scientific_journal-610 1 39 169 169 NUM american_scientific_journal-610 1 40 β„š β„š NOUN american_scientific_journal-610 1 41 is be AUX american_scientific_journal-610 1 42 a a DET american_scientific_journal-610 1 43 convergence convergence NOUN american_scientific_journal-610 1 44 set set VERB american_scientific_journal-610 1 45 basma basma PROPN american_scientific_journal-610 1 46 al al PROPN american_scientific_journal-610 1 47 - - PUNCT american_scientific_journal-610 1 48 shutnawia shutnawia NOUN american_scientific_journal-610 1 49 * * PROPN american_scientific_journal-610 1 50 , , PUNCT american_scientific_journal-610 1 51 mohammad mohammad PROPN american_scientific_journal-610 1 52 zannonb zannonb PROPN american_scientific_journal-610 1 53 a a DET american_scientific_journal-610 1 54 department department NOUN american_scientific_journal-610 1 55 of of ADP american_scientific_journal-610 1 56 mathematics mathematic NOUN american_scientific_journal-610 1 57 , , PUNCT american_scientific_journal-610 1 58 tafila tafila NOUN american_scientific_journal-610 1 59 technical technical ADJ american_scientific_journal-610 1 60 university university PROPN american_scientific_journal-610 1 61 , , PUNCT american_scientific_journal-610 1 62 p.o p.o PROPN american_scientific_journal-610 1 63 . . PROPN american_scientific_journal-610 1 64 box box PROPN american_scientific_journal-610 1 65 179 179 NUM american_scientific_journal-610 1 66 , , PUNCT american_scientific_journal-610 1 67 tafila tafila NOUN american_scientific_journal-610 1 68 66110 66110 NUM american_scientific_journal-610 1 69 , , PUNCT american_scientific_journal-610 1 70 jordan jordan PROPN american_scientific_journal-610 1 71 . . PUNCT american_scientific_journal-610 2 1 b b PROPN american_scientific_journal-610 2 2 department department PROPN american_scientific_journal-610 2 3 of of ADP american_scientific_journal-610 2 4 mathematics mathematic NOUN american_scientific_journal-610 2 5 , , PUNCT american_scientific_journal-610 2 6 tafila tafila NOUN american_scientific_journal-610 2 7 technical technical ADJ american_scientific_journal-610 2 8 university university PROPN american_scientific_journal-610 2 9 , , PUNCT american_scientific_journal-610 2 10 p.o p.o PROPN american_scientific_journal-610 2 11 . . PROPN american_scientific_journal-610 2 12 box179 box179 PROPN american_scientific_journal-610 2 13 , , PUNCT american_scientific_journal-610 2 14 tafila tafila NOUN american_scientific_journal-610 2 15 66110 66110 NUM american_scientific_journal-610 2 16 , , PUNCT american_scientific_journal-610 2 17 jordan jordan PROPN american_scientific_journal-610 2 18 . . PUNCT american_scientific_journal-610 3 1 a a DET american_scientific_journal-610 3 2 basma@ttu.edu.jo basma@ttu.edu.jo NOUN american_scientific_journal-610 3 3 bzanno1ms@gmail.com bzanno1ms@gmail.com NOUN american_scientific_journal-610 3 4 abstract abstract NOUN american_scientific_journal-610 3 5 in in ADP american_scientific_journal-610 3 6 this this DET american_scientific_journal-610 3 7 paper paper NOUN american_scientific_journal-610 3 8 we we PRON american_scientific_journal-610 3 9 consider consider VERB american_scientific_journal-610 3 10 the the DET american_scientific_journal-610 3 11 convergence convergence NOUN american_scientific_journal-610 3 12 sets set NOUN american_scientific_journal-610 3 13 of of ADP american_scientific_journal-610 3 14 formal formal ADJ american_scientific_journal-610 3 15 power power NOUN american_scientific_journal-610 3 16 series series NOUN american_scientific_journal-610 3 17 of of ADP american_scientific_journal-610 3 18 the the DET american_scientific_journal-610 3 19 form form NOUN american_scientific_journal-610 3 20 𝑓(𝑧 𝑓(𝑧 PROPN american_scientific_journal-610 3 21 , , PUNCT american_scientific_journal-610 3 22 𝑑 𝑑 NOUN american_scientific_journal-610 3 23 ) ) PUNCT american_scientific_journal-610 3 24 = = SYM american_scientific_journal-610 3 25 βˆ‘ βˆ‘ PUNCT american_scientific_journal-610 3 26 𝑓𝑗(𝑧)π‘‘π‘—βˆž 𝑓𝑗(𝑧)π‘‘π‘—βˆž PROPN american_scientific_journal-610 3 27 𝑗=0 𝑗=0 X american_scientific_journal-610 3 28 , , PUNCT american_scientific_journal-610 3 29 where where SCONJ american_scientific_journal-610 3 30 𝑓𝑗(𝑧 𝑓𝑗(𝑧 NUM american_scientific_journal-610 3 31 ) ) PUNCT american_scientific_journal-610 3 32 are be AUX american_scientific_journal-610 3 33 polynomials polynomial NOUN american_scientific_journal-610 3 34 functions function NOUN american_scientific_journal-610 3 35 on on ADP american_scientific_journal-610 3 36 a a DET american_scientific_journal-610 3 37 domain domain NOUN american_scientific_journal-610 3 38 Ο‰ Ο‰ NOUN american_scientific_journal-610 3 39 in in ADP american_scientific_journal-610 3 40 β„‚. β„‚. PROPN american_scientific_journal-610 3 41 a a DET american_scientific_journal-610 3 42 subset subset ADJ american_scientific_journal-610 3 43 𝐸 𝐸 PROPN american_scientific_journal-610 3 44 of of ADP american_scientific_journal-610 3 45 Ο‰ Ο‰ PROPN american_scientific_journal-610 3 46 is be AUX american_scientific_journal-610 3 47 said say VERB american_scientific_journal-610 3 48 to to PART american_scientific_journal-610 3 49 be be AUX american_scientific_journal-610 3 50 convergence convergence NOUN american_scientific_journal-610 3 51 set set VERB american_scientific_journal-610 3 52 if if SCONJ american_scientific_journal-610 3 53 there there PRON american_scientific_journal-610 3 54 is be VERB american_scientific_journal-610 3 55 a a DET american_scientific_journal-610 3 56 series series NOUN american_scientific_journal-610 3 57 𝑓(𝑧 𝑓(𝑧 PROPN american_scientific_journal-610 3 58 , , PUNCT american_scientific_journal-610 3 59 𝑑 𝑑 NOUN american_scientific_journal-610 3 60 ) ) PUNCT american_scientific_journal-610 3 61 such such ADJ american_scientific_journal-610 3 62 that that SCONJ american_scientific_journal-610 3 63 𝐸 𝐸 PROPN american_scientific_journal-610 3 64 is be AUX american_scientific_journal-610 3 65 exactly exactly ADV american_scientific_journal-610 3 66 the the DET american_scientific_journal-610 3 67 set set NOUN american_scientific_journal-610 3 68 of of ADP american_scientific_journal-610 3 69 points point NOUN american_scientific_journal-610 3 70 𝑧 𝑧 VERB american_scientific_journal-610 3 71 for for ADP american_scientific_journal-610 3 72 which which PRON american_scientific_journal-610 3 73 𝑓(𝑧 𝑓(𝑧 PROPN american_scientific_journal-610 3 74 , , PUNCT american_scientific_journal-610 3 75 𝑑 𝑑 NOUN american_scientific_journal-610 3 76 ) ) PUNCT american_scientific_journal-610 3 77 converges converge NOUN american_scientific_journal-610 3 78 as as ADP american_scientific_journal-610 3 79 a a DET american_scientific_journal-610 3 80 power power NOUN american_scientific_journal-610 3 81 series series NOUN american_scientific_journal-610 3 82 in in ADP american_scientific_journal-610 3 83 t t PROPN american_scientific_journal-610 3 84 in in ADP american_scientific_journal-610 3 85 some some DET american_scientific_journal-610 3 86 neighborhood neighborhood NOUN american_scientific_journal-610 3 87 of of ADP american_scientific_journal-610 3 88 the the DET american_scientific_journal-610 3 89 origin origin NOUN american_scientific_journal-610 3 90 . . PUNCT american_scientific_journal-610 4 1 we we PRON american_scientific_journal-610 4 2 prove prove VERB american_scientific_journal-610 4 3 that that SCONJ american_scientific_journal-610 4 4 β„š β„š PROPN american_scientific_journal-610 4 5 is be AUX american_scientific_journal-610 4 6 a a DET american_scientific_journal-610 4 7 convergence convergence NOUN american_scientific_journal-610 4 8 set set VERB american_scientific_journal-610 4 9 . . PUNCT american_scientific_journal-610 5 1 keywords keyword NOUN american_scientific_journal-610 5 2 : : PUNCT american_scientific_journal-610 5 3 formal formal ADJ american_scientific_journal-610 5 4 power power NOUN american_scientific_journal-610 5 5 series series NOUN american_scientific_journal-610 5 6 ; ; PUNCT american_scientific_journal-610 5 7 convergence convergence NOUN american_scientific_journal-610 5 8 sets set NOUN american_scientific_journal-610 5 9 , , PUNCT american_scientific_journal-610 5 10 β„š β„š PROPN american_scientific_journal-610 5 11 the the DET american_scientific_journal-610 5 12 set set NOUN american_scientific_journal-610 5 13 of of ADP american_scientific_journal-610 5 14 rational rational ADJ american_scientific_journal-610 5 15 numbers number NOUN american_scientific_journal-610 5 16 , , PUNCT american_scientific_journal-610 5 17 quasi quasi ADJ american_scientific_journal-610 5 18 - - ADJ american_scientific_journal-610 5 19 simply simply ADV american_scientific_journal-610 5 20 - - PUNCT american_scientific_journal-610 5 21 connected connect VERB american_scientific_journal-610 5 22 sets set NOUN american_scientific_journal-610 5 23 . . PUNCT american_scientific_journal-610 6 1 1 1 X american_scientific_journal-610 6 2 . . X american_scientific_journal-610 6 3 introduction introduction NOUN american_scientific_journal-610 6 4 let let VERB american_scientific_journal-610 6 5 β„‚[z β„‚[z NOUN american_scientific_journal-610 6 6 ] ] PUNCT american_scientific_journal-610 6 7 be be AUX american_scientific_journal-610 6 8 the the DET american_scientific_journal-610 6 9 set set NOUN american_scientific_journal-610 6 10 of of ADP american_scientific_journal-610 6 11 polynomials polynomial NOUN american_scientific_journal-610 6 12 in in ADP american_scientific_journal-610 6 13 z z PROPN american_scientific_journal-610 6 14 , , PUNCT american_scientific_journal-610 6 15 β„‚[[𝑧 β„‚[[𝑧 PROPN american_scientific_journal-610 6 16 ] ] PUNCT american_scientific_journal-610 6 17 ] ] PUNCT american_scientific_journal-610 6 18 be be VERB american_scientific_journal-610 6 19 the the DET american_scientific_journal-610 6 20 set set NOUN american_scientific_journal-610 6 21 of of ADP american_scientific_journal-610 6 22 formal formal ADJ american_scientific_journal-610 6 23 power power NOUN american_scientific_journal-610 6 24 series series NOUN american_scientific_journal-610 6 25 , , PUNCT american_scientific_journal-610 6 26 and and CCONJ american_scientific_journal-610 6 27 β„‚[𝑧][[𝑑 β„‚[𝑧][[𝑑 NOUN american_scientific_journal-610 6 28 ] ] X american_scientific_journal-610 6 29 ] ] PUNCT american_scientific_journal-610 6 30 be be VERB american_scientific_journal-610 6 31 the the DET american_scientific_journal-610 6 32 set set NOUN american_scientific_journal-610 6 33 of of ADP american_scientific_journal-610 6 34 formal formal ADJ american_scientific_journal-610 6 35 power power NOUN american_scientific_journal-610 6 36 series series NOUN american_scientific_journal-610 6 37 in in ADP american_scientific_journal-610 6 38 𝑑 𝑑 PROPN american_scientific_journal-610 6 39 with with ADP american_scientific_journal-610 6 40 coefficients coefficient NOUN american_scientific_journal-610 6 41 being be AUX american_scientific_journal-610 6 42 polynomials polynomial NOUN american_scientific_journal-610 6 43 in in ADP american_scientific_journal-610 6 44 𝑧. 𝑧. NOUN american_scientific_journal-610 6 45 in in ADP american_scientific_journal-610 6 46 my my PRON american_scientific_journal-610 6 47 dissertation dissertation NOUN american_scientific_journal-610 7 1 [ [ X american_scientific_journal-610 7 2 3 3 X american_scientific_journal-610 7 3 ] ] PUNCT american_scientific_journal-610 7 4 i i PRON american_scientific_journal-610 7 5 considered consider VERB american_scientific_journal-610 7 6 the the DET american_scientific_journal-610 7 7 formal formal ADJ american_scientific_journal-610 7 8 power power NOUN american_scientific_journal-610 7 9 series series PROPN american_scientific_journal-610 7 10 𝒇(𝒛 𝒇(𝒛 PROPN american_scientific_journal-610 7 11 ) ) PUNCT american_scientific_journal-610 7 12 = = PUNCT american_scientific_journal-610 7 13 π’‚πŸŽ π’‚πŸŽ PROPN american_scientific_journal-610 7 14 + + CCONJ american_scientific_journal-610 7 15 βˆ‘ βˆ‘ PUNCT american_scientific_journal-610 7 16 π’‚πœΆπŸπ’› π’‚πœΆπŸπ’› VERB american_scientific_journal-610 7 17 𝜢𝟏|𝜢𝟏|=𝟏 𝜢𝟏|𝜢𝟏|=𝟏 NOUN american_scientific_journal-610 7 18 + + CCONJ american_scientific_journal-610 7 19 βˆ‘ βˆ‘ PROPN american_scientific_journal-610 7 20 π’‚πœΆπŸπ’› π’‚πœΆπŸπ’› NOUN american_scientific_journal-610 7 21 𝜢𝟐|𝜢𝟐|=𝟐 𝜢𝟐|𝜢𝟐|=𝟐 NUM american_scientific_journal-610 7 22 + + NUM american_scientific_journal-610 7 23 β‹―+ β‹―+ NOUN american_scientific_journal-610 7 24 βˆ‘ βˆ‘ ADV american_scientific_journal-610 7 25 π’‚πœΆπ’π’› π’‚πœΆπ’π’› NOUN american_scientific_journal-610 7 26 πœΆπ’|πœΆπ’|=𝒏 πœΆπ’|πœΆπ’|=𝒏 NUM american_scientific_journal-610 7 27 , , PUNCT american_scientific_journal-610 7 28 where where SCONJ american_scientific_journal-610 7 29 𝛼𝑗= 𝛼𝑗= NOUN american_scientific_journal-610 7 30 ( ( PUNCT american_scientific_journal-610 7 31 𝛼𝑗1 𝛼𝑗1 PROPN american_scientific_journal-610 7 32 , , PUNCT american_scientific_journal-610 7 33 … … NUM american_scientific_journal-610 7 34 , , PUNCT american_scientific_journal-610 7 35 𝛼𝑗𝑛 𝛼𝑗𝑛 NOUN american_scientific_journal-610 7 36 ) ) PUNCT american_scientific_journal-610 7 37 are be AUX american_scientific_journal-610 7 38 the the DET american_scientific_journal-610 7 39 n n CCONJ american_scientific_journal-610 7 40 - - PUNCT american_scientific_journal-610 7 41 multiple multiple ADJ american_scientific_journal-610 7 42 index index NOUN american_scientific_journal-610 7 43 and and CCONJ american_scientific_journal-610 7 44 |𝛼𝑗 |𝛼𝑗 NOUN american_scientific_journal-610 7 45 | | ADV american_scientific_journal-610 7 46 = = PUNCT american_scientific_journal-610 7 47 |𝛼𝑗1| |𝛼𝑗1| PROPN american_scientific_journal-610 7 48 + + NUM american_scientific_journal-610 7 49 β‹―+ β‹―+ NOUN american_scientific_journal-610 7 50 οΏ½ οΏ½ PROPN american_scientific_journal-610 7 51 𝛼𝑗𝑛 𝛼𝑗𝑛 PROPN american_scientific_journal-610 7 52 οΏ½ οΏ½ PROPN american_scientific_journal-610 7 53 . . PUNCT american_scientific_journal-610 8 1 many many ADJ american_scientific_journal-610 8 2 research research NOUN american_scientific_journal-610 8 3 concerning concern VERB american_scientific_journal-610 8 4 convergence convergence NOUN american_scientific_journal-610 8 5 ( ( PUNCT american_scientific_journal-610 8 6 or or CCONJ american_scientific_journal-610 8 7 formal formal ADJ american_scientific_journal-610 8 8 ) ) PUNCT american_scientific_journal-610 8 9 series series NOUN american_scientific_journal-610 8 10 [ [ X american_scientific_journal-610 8 11 7 7 NUM american_scientific_journal-610 8 12 , , PUNCT american_scientific_journal-610 8 13 1 1 NUM american_scientific_journal-610 8 14 , , PUNCT american_scientific_journal-610 8 15 11 11 NUM american_scientific_journal-610 8 16 , , PUNCT american_scientific_journal-610 8 17 8 8 NUM american_scientific_journal-610 8 18 , , PUNCT american_scientific_journal-610 8 19 5 5 NUM american_scientific_journal-610 8 20 ] ] PUNCT american_scientific_journal-610 8 21 . . PUNCT american_scientific_journal-610 9 1 the the DET american_scientific_journal-610 9 2 general general ADJ american_scientific_journal-610 9 3 description description NOUN american_scientific_journal-610 9 4 of of ADP american_scientific_journal-610 9 5 these these DET american_scientific_journal-610 9 6 problems problem NOUN american_scientific_journal-610 9 7 is be AUX american_scientific_journal-610 9 8 given give VERB american_scientific_journal-610 9 9 in in ADP american_scientific_journal-610 9 10 [ [ X american_scientific_journal-610 9 11 12 12 NUM american_scientific_journal-610 9 12 ] ] PUNCT american_scientific_journal-610 9 13 . . PUNCT american_scientific_journal-610 10 1 recently recently ADV american_scientific_journal-610 10 2 there there PRON american_scientific_journal-610 10 3 were be VERB american_scientific_journal-610 10 4 new new ADJ american_scientific_journal-610 10 5 researches research NOUN american_scientific_journal-610 10 6 concerning concern VERB american_scientific_journal-610 10 7 the the DET american_scientific_journal-610 10 8 power power NOUN american_scientific_journal-610 10 9 series series NOUN american_scientific_journal-610 10 10 when when SCONJ american_scientific_journal-610 10 11 the the DET american_scientific_journal-610 10 12 coefficients coefficient NOUN american_scientific_journal-610 10 13 are be AUX american_scientific_journal-610 10 14 polynomials polynomial NOUN american_scientific_journal-610 10 15 of of ADP american_scientific_journal-610 10 16 two two NUM american_scientific_journal-610 10 17 or or CCONJ american_scientific_journal-610 10 18 more more ADJ american_scientific_journal-610 10 19 complex complex ADJ american_scientific_journal-610 10 20 variables variable NOUN american_scientific_journal-610 10 21 [ [ X american_scientific_journal-610 10 22 2 2 NUM american_scientific_journal-610 10 23 , , PUNCT american_scientific_journal-610 10 24 4 4 NUM american_scientific_journal-610 10 25 , , PUNCT american_scientific_journal-610 10 26 9 9 NUM american_scientific_journal-610 10 27 , , PUNCT american_scientific_journal-610 10 28 6 6 NUM american_scientific_journal-610 10 29 ] ] PUNCT american_scientific_journal-610 10 30 . . PUNCT american_scientific_journal-610 11 1 consider consider VERB american_scientific_journal-610 11 2 𝐹(𝑧 𝐹(𝑧 NUM american_scientific_journal-610 11 3 , , PUNCT american_scientific_journal-610 11 4 𝑑 𝑑 NOUN american_scientific_journal-610 11 5 ) ) PUNCT american_scientific_journal-610 11 6 = = PUNCT american_scientific_journal-610 12 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-610 12 2 π‘ƒπ‘š(𝑧)π‘‘π‘šβˆž π‘ƒπ‘š(𝑧)π‘‘π‘šβˆž PROPN american_scientific_journal-610 12 3 π‘š=0 π‘š=0 PUNCT american_scientific_journal-610 12 4 . . PUNCT american_scientific_journal-610 13 1 suppose suppose VERB american_scientific_journal-610 13 2 that that SCONJ american_scientific_journal-610 13 3 𝐹(𝑧 𝐹(𝑧 ADP american_scientific_journal-610 13 4 , , PUNCT american_scientific_journal-610 13 5 𝑑 𝑑 NOUN american_scientific_journal-610 13 6 ) ) PUNCT american_scientific_journal-610 13 7 as as ADP american_scientific_journal-610 13 8 a a DET american_scientific_journal-610 13 9 power power NOUN american_scientific_journal-610 13 10 series series NOUN american_scientific_journal-610 13 11 in in ADP american_scientific_journal-610 13 12 𝑑 𝑑 PROPN american_scientific_journal-610 13 13 converges converge NOUN american_scientific_journal-610 13 14 for for ADP american_scientific_journal-610 13 15 𝑧 𝑧 PRON american_scientific_journal-610 13 16 in in ADP american_scientific_journal-610 13 17 a a DET american_scientific_journal-610 13 18 set set NOUN american_scientific_journal-610 13 19 𝐸 𝐸 PROPN american_scientific_journal-610 13 20 βŠ‚ βŠ‚ PROPN american_scientific_journal-610 13 21 ℂ𝑛. ℂ𝑛. PROPN american_scientific_journal-610 14 1 i i PRON american_scientific_journal-610 14 2 mentioned mention VERB american_scientific_journal-610 14 3 in in ADP american_scientific_journal-610 14 4 my my PRON american_scientific_journal-610 14 5 dissertation dissertation NOUN american_scientific_journal-610 14 6 that that SCONJ american_scientific_journal-610 14 7 the the DET american_scientific_journal-610 14 8 set set VERB american_scientific_journal-610 14 9 β„š β„š PROPN american_scientific_journal-610 14 10 of of ADP american_scientific_journal-610 14 11 rational rational ADJ american_scientific_journal-610 14 12 numbers number NOUN american_scientific_journal-610 14 13 is be AUX american_scientific_journal-610 14 14 a a DET american_scientific_journal-610 14 15 convergence convergence NOUN american_scientific_journal-610 14 16 set set VERB american_scientific_journal-610 14 17 in in ADP american_scientific_journal-610 14 18 this this DET american_scientific_journal-610 14 19 paper paper NOUN american_scientific_journal-610 14 20 we we PRON american_scientific_journal-610 14 21 prove prove VERB american_scientific_journal-610 14 22 this this DET american_scientific_journal-610 14 23 corollary corollary NOUN american_scientific_journal-610 14 24 . . PUNCT american_scientific_journal-610 15 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-610 16 1 * * PUNCT american_scientific_journal-610 16 2 corresponding correspond VERB american_scientific_journal-610 16 3 author author NOUN american_scientific_journal-610 16 4 . . PUNCT american_scientific_journal-610 17 1 e e X american_scientific_journal-610 17 2 - - NOUN american_scientific_journal-610 17 3 mail mail NOUN american_scientific_journal-610 17 4 address address NOUN american_scientific_journal-610 17 5 : : PUNCT american_scientific_journal-610 17 6 basma@ttu.edu.jo basma@ttu.edu.jo NOUN american_scientific_journal-610 17 7 . . PUNCT american_scientific_journal-610 18 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-610 18 2 american american PROPN american_scientific_journal-610 18 3 scientific scientific PROPN american_scientific_journal-610 18 4 research research NOUN american_scientific_journal-610 18 5 journal journal NOUN american_scientific_journal-610 18 6 for for ADP american_scientific_journal-610 18 7 engineering engineering NOUN american_scientific_journal-610 18 8 , , PUNCT american_scientific_journal-610 18 9 technology technology NOUN american_scientific_journal-610 18 10 , , PUNCT american_scientific_journal-610 18 11 and and CCONJ american_scientific_journal-610 18 12 sciences science NOUN american_scientific_journal-610 18 13 ( ( PUNCT american_scientific_journal-610 18 14 asrjets asrjet NOUN american_scientific_journal-610 18 15 ) ) PUNCT american_scientific_journal-610 18 16 ( ( PUNCT american_scientific_journal-610 18 17 2015 2015 NUM american_scientific_journal-610 18 18 ) ) PUNCT american_scientific_journal-610 18 19 volume volume NOUN american_scientific_journal-610 18 20 11 11 NUM american_scientific_journal-610 18 21 , , PUNCT american_scientific_journal-610 18 22 no no DET american_scientific_journal-610 18 23 1 1 NUM american_scientific_journal-610 18 24 , , PUNCT american_scientific_journal-610 18 25 pp pp ADV american_scientific_journal-610 18 26 169 169 NUM american_scientific_journal-610 18 27 - - SYM american_scientific_journal-610 18 28 172 172 NUM american_scientific_journal-610 18 29 2 2 NUM american_scientific_journal-610 18 30 . . X american_scientific_journal-610 19 1 β„š β„š PROPN american_scientific_journal-610 19 2 is be AUX american_scientific_journal-610 19 3 a a DET american_scientific_journal-610 19 4 convergence convergence NOUN american_scientific_journal-610 19 5 sets set VERB american_scientific_journal-610 19 6 2.1 2.1 NUM american_scientific_journal-610 19 7 . . PUNCT american_scientific_journal-610 20 1 definition definition NOUN american_scientific_journal-610 20 2 [ [ X american_scientific_journal-610 20 3 3 3 X american_scientific_journal-610 20 4 ] ] X american_scientific_journal-610 20 5 a a DET american_scientific_journal-610 20 6 power power NOUN american_scientific_journal-610 20 7 series series NOUN american_scientific_journal-610 20 8 𝑓 𝑓 PROPN american_scientific_journal-610 20 9 ∈ ∈ PROPN american_scientific_journal-610 20 10 β„‚[[𝑧1 β„‚[[𝑧1 PROPN american_scientific_journal-610 20 11 , , PUNCT american_scientific_journal-610 20 12 … … PUNCT american_scientific_journal-610 20 13 , , PUNCT american_scientific_journal-610 20 14 𝑧𝑛 𝑧𝑛 ADP american_scientific_journal-610 20 15 ] ] X american_scientific_journal-610 20 16 ] ] X american_scientific_journal-610 20 17 is be AUX american_scientific_journal-610 20 18 said say VERB american_scientific_journal-610 20 19 to to PART american_scientific_journal-610 20 20 be be AUX american_scientific_journal-610 20 21 convergent convergent ADJ american_scientific_journal-610 20 22 if if SCONJ american_scientific_journal-610 20 23 there there PRON american_scientific_journal-610 20 24 is be VERB american_scientific_journal-610 20 25 a a DET american_scientific_journal-610 20 26 constant constant ADJ american_scientific_journal-610 20 27 c c NOUN american_scientific_journal-610 20 28 such such ADJ american_scientific_journal-610 20 29 that that DET american_scientific_journal-610 20 30 |π‘Žπ‘˜1, |π‘Žπ‘˜1, NOUN american_scientific_journal-610 20 31 … … SYM american_scientific_journal-610 20 32 ,π‘˜π‘›| ,π‘˜π‘›| PUNCT american_scientific_journal-610 20 33 ≀ ≀ ADJ american_scientific_journal-610 20 34 πΆπ‘˜1+β‹―+π‘˜π‘› πΆπ‘˜1+β‹―+π‘˜π‘› PROPN american_scientific_journal-610 20 35 for for ADP american_scientific_journal-610 20 36 all all DET american_scientific_journal-610 20 37 ( ( PUNCT american_scientific_journal-610 20 38 π‘˜1 π‘˜1 PROPN american_scientific_journal-610 20 39 , , PUNCT american_scientific_journal-610 20 40 … … PUNCT american_scientific_journal-610 20 41 , , PUNCT american_scientific_journal-610 20 42 π‘˜π‘› π‘˜π‘› PROPN american_scientific_journal-610 20 43 ) ) PUNCT american_scientific_journal-610 20 44 β‰  β‰  PROPN american_scientific_journal-610 20 45 ( ( PUNCT american_scientific_journal-610 20 46 0 0 NUM american_scientific_journal-610 20 47 , , PUNCT american_scientific_journal-610 20 48 … … PUNCT american_scientific_journal-610 20 49 , , PUNCT american_scientific_journal-610 20 50 0 0 NUM american_scientific_journal-610 20 51 ) ) PUNCT american_scientific_journal-610 20 52 . . PUNCT american_scientific_journal-610 21 1 a a DET american_scientific_journal-610 21 2 power power NOUN american_scientific_journal-610 21 3 series series NOUN american_scientific_journal-610 21 4 𝑓 𝑓 NOUN american_scientific_journal-610 21 5 is be AUX american_scientific_journal-610 21 6 said say VERB american_scientific_journal-610 21 7 to to PART american_scientific_journal-610 21 8 be be AUX american_scientific_journal-610 21 9 divergent divergent ADJ american_scientific_journal-610 21 10 if if SCONJ american_scientific_journal-610 21 11 it it PRON american_scientific_journal-610 21 12 is be AUX american_scientific_journal-610 21 13 not not PART american_scientific_journal-610 21 14 convergent convergent ADJ american_scientific_journal-610 21 15 . . PUNCT american_scientific_journal-610 22 1 2.2 2.2 NUM american_scientific_journal-610 22 2 . . PUNCT american_scientific_journal-610 22 3 definition definition NOUN american_scientific_journal-610 22 4 [ [ X american_scientific_journal-610 22 5 3 3 X american_scientific_journal-610 22 6 ] ] X american_scientific_journal-610 22 7 let let VERB american_scientific_journal-610 22 8 𝑓(𝑧 𝑓(𝑧 NUM american_scientific_journal-610 22 9 , , PUNCT american_scientific_journal-610 22 10 𝑑 𝑑 NOUN american_scientific_journal-610 22 11 ) ) PUNCT american_scientific_journal-610 22 12 ∈ ∈ PROPN american_scientific_journal-610 22 13 β„‚[𝑧][[𝑑 β„‚[𝑧][[𝑑 PROPN american_scientific_journal-610 22 14 ] ] X american_scientific_journal-610 22 15 ] ] PUNCT american_scientific_journal-610 22 16 . . PUNCT american_scientific_journal-610 23 1 define define VERB american_scientific_journal-610 23 2 the the DET american_scientific_journal-610 23 3 convergence convergence NOUN american_scientific_journal-610 23 4 set set VERB american_scientific_journal-610 23 5 of of ADP american_scientific_journal-610 23 6 𝑓 𝑓 PRON american_scientific_journal-610 23 7 to to PART american_scientific_journal-610 23 8 be be AUX american_scientific_journal-610 23 9 conv conv ADJ american_scientific_journal-610 23 10 ( ( PUNCT american_scientific_journal-610 23 11 f f X american_scientific_journal-610 23 12 ) ) PUNCT american_scientific_journal-610 23 13 = = PRON american_scientific_journal-610 24 1 { { PUNCT american_scientific_journal-610 24 2 z z NOUN american_scientific_journal-610 24 3 ∈ ∈ PROPN american_scientific_journal-610 24 4 β„‚ β„‚ PROPN american_scientific_journal-610 24 5 : : PUNCT american_scientific_journal-610 24 6 f(z f(z PROPN american_scientific_journal-610 24 7 , , PUNCT american_scientific_journal-610 24 8 t t PROPN american_scientific_journal-610 24 9 ) ) PUNCT american_scientific_journal-610 24 10 converges converge NOUN american_scientific_journal-610 24 11 in in ADP american_scientific_journal-610 24 12 t t PROPN american_scientific_journal-610 24 13 } } PUNCT american_scientific_journal-610 24 14 . . PUNCT american_scientific_journal-610 25 1 2.3 2.3 NUM american_scientific_journal-610 25 2 . . PUNCT american_scientific_journal-610 25 3 definition definition NOUN american_scientific_journal-610 25 4 [ [ X american_scientific_journal-610 25 5 3 3 X american_scientific_journal-610 25 6 ] ] PUNCT american_scientific_journal-610 25 7 a a DET american_scientific_journal-610 25 8 subset subset NOUN american_scientific_journal-610 25 9 𝐸 𝐸 PROPN american_scientific_journal-610 25 10 ∈ ∈ PROPN american_scientific_journal-610 25 11 β„‚ β„‚ PROPN american_scientific_journal-610 25 12 is be AUX american_scientific_journal-610 25 13 said say VERB american_scientific_journal-610 25 14 to to PART american_scientific_journal-610 25 15 be be AUX american_scientific_journal-610 25 16 a a DET american_scientific_journal-610 25 17 convergence convergence NOUN american_scientific_journal-610 25 18 set set VERB american_scientific_journal-610 25 19 if if SCONJ american_scientific_journal-610 25 20 there there PRON american_scientific_journal-610 25 21 exists exist VERB american_scientific_journal-610 25 22 an an DET american_scientific_journal-610 25 23 𝑓 𝑓 PRON american_scientific_journal-610 25 24 ∈ ∈ PROPN american_scientific_journal-610 25 25 β„‚[𝑧][[𝑑 β„‚[𝑧][[𝑑 NOUN american_scientific_journal-610 25 26 ] ] X american_scientific_journal-610 25 27 ] ] X american_scientific_journal-610 25 28 such such ADJ american_scientific_journal-610 25 29 that that SCONJ american_scientific_journal-610 25 30 e e NOUN american_scientific_journal-610 25 31 = = NOUN american_scientific_journal-610 25 32 conv(f conv(f PROPN american_scientific_journal-610 25 33 ) ) PUNCT american_scientific_journal-610 25 34 . . PUNCT american_scientific_journal-610 26 1 moreover moreover ADV american_scientific_journal-610 26 2 we we PRON american_scientific_journal-610 26 3 proved prove VERB american_scientific_journal-610 26 4 the the DET american_scientific_journal-610 26 5 following follow VERB american_scientific_journal-610 26 6 theorem theorem NOUN american_scientific_journal-610 26 7 , , PUNCT american_scientific_journal-610 26 8 2.4 2.4 NUM american_scientific_journal-610 26 9 . . PUNCT american_scientific_journal-610 27 1 theorem theorem VERB american_scientific_journal-610 27 2 [ [ X american_scientific_journal-610 27 3 3 3 NUM american_scientific_journal-610 27 4 ] ] PUNCT american_scientific_journal-610 27 5 let let VERB american_scientific_journal-610 27 6 𝑆 𝑆 PROPN american_scientific_journal-610 27 7 = = PRON american_scientific_journal-610 27 8 { { PUNCT american_scientific_journal-610 27 9 𝑧1 𝑧1 PROPN american_scientific_journal-610 27 10 , , PUNCT american_scientific_journal-610 27 11 𝑧2 𝑧2 PROPN american_scientific_journal-610 27 12 , , PUNCT american_scientific_journal-610 27 13 … … PUNCT american_scientific_journal-610 27 14 } } PUNCT american_scientific_journal-610 27 15 be be AUX american_scientific_journal-610 27 16 a a DET american_scientific_journal-610 27 17 countable countable ADJ american_scientific_journal-610 27 18 infinite infinite ADJ american_scientific_journal-610 27 19 subset subset NOUN american_scientific_journal-610 27 20 of of ADP american_scientific_journal-610 27 21 β„‚. β„‚. NOUN american_scientific_journal-610 27 22 define define VERB american_scientific_journal-610 27 23 an an DET american_scientific_journal-610 27 24 f f PROPN american_scientific_journal-610 27 25 ∈ ∈ PROPN american_scientific_journal-610 27 26 β„‚[z][[t β„‚[z][[t NOUN american_scientific_journal-610 27 27 ] ] X american_scientific_journal-610 27 28 ] ] PUNCT american_scientific_journal-610 27 29 by by ADP american_scientific_journal-610 27 30 𝑭(𝒛 𝑭(𝒛 PROPN american_scientific_journal-610 27 31 , , PUNCT american_scientific_journal-610 27 32 𝒕 𝒕 NOUN american_scientific_journal-610 27 33 ) ) PUNCT american_scientific_journal-610 27 34 = = SYM american_scientific_journal-610 27 35 οΏ½ οΏ½ PROPN american_scientific_journal-610 27 36 π‘ͺ𝒏 π‘ͺ𝒏 PROPN american_scientific_journal-610 27 37 [ [ PUNCT american_scientific_journal-610 27 38 οΏ½ οΏ½ PROPN american_scientific_journal-610 27 39 ( ( PUNCT american_scientific_journal-610 27 40 𝒛 𝒛 NOUN american_scientific_journal-610 27 41 βˆ’ βˆ’ PROPN american_scientific_journal-610 27 42 𝒛𝒋 𝒛𝒋 NOUN american_scientific_journal-610 27 43 ) ) PUNCT american_scientific_journal-610 27 44 ] ] PUNCT american_scientific_journal-610 28 1 𝒏 𝒏 PROPN american_scientific_journal-610 28 2 𝒋=𝟏 𝒋=𝟏 PUNCT american_scientific_journal-610 28 3 ∞ ∞ PROPN american_scientific_journal-610 28 4 𝒏=𝟎 𝒏=𝟎 PUNCT american_scientific_journal-610 28 5 𝒕𝒏 𝒕𝒏 NOUN american_scientific_journal-610 28 6 , , PUNCT american_scientific_journal-610 28 7 where where SCONJ american_scientific_journal-610 28 8 𝐢𝑛 𝐢𝑛 NOUN american_scientific_journal-610 28 9 = = X american_scientific_journal-610 28 10 ( ( PUNCT american_scientific_journal-610 28 11 𝑛 𝑛 PROPN american_scientific_journal-610 28 12 𝛾𝑛 𝛾𝑛 INTJ american_scientific_journal-610 28 13 οΏ½ οΏ½ PROPN american_scientific_journal-610 28 14 ) ) PUNCT american_scientific_journal-610 28 15 𝑛 𝑛 PROPN american_scientific_journal-610 28 16 , , PUNCT american_scientific_journal-610 28 17 and and CCONJ american_scientific_journal-610 28 18 πœΈπ’ πœΈπ’ X american_scientific_journal-610 28 19 = = PROPN american_scientific_journal-610 28 20 𝐦𝐒𝐧 𝐦𝐒𝐧 PROPN american_scientific_journal-610 28 21 οΏ½ οΏ½ PROPN american_scientific_journal-610 28 22 𝟏 𝟏 NUM american_scientific_journal-610 28 23 𝟐 𝟐 NUM american_scientific_journal-610 28 24 𝐦𝐒𝐧 𝐦𝐒𝐧 NOUN american_scientific_journal-610 28 25 πŸβ‰€π’‹β‰€π’+𝟏 πŸβ‰€π’‹β‰€π’+𝟏 X american_scientific_journal-610 28 26 οΏ½ οΏ½ PROPN american_scientific_journal-610 28 27 π’›π’Š π’›π’Š PRON american_scientific_journal-610 28 28 βˆ’ βˆ’ PROPN american_scientific_journal-610 28 29 𝒛𝒋 𝒛𝒋 NOUN american_scientific_journal-610 28 30 οΏ½ οΏ½ PROPN american_scientific_journal-610 28 31 , , PUNCT american_scientific_journal-610 28 32 𝟏 𝟏 NUM american_scientific_journal-610 28 33 𝒏⁄ 𝒏⁄ PROPN american_scientific_journal-610 28 34 οΏ½ οΏ½ PROPN american_scientific_journal-610 28 35 . . PUNCT american_scientific_journal-610 29 1 then then ADV american_scientific_journal-610 29 2 conv(f conv(f ADP american_scientific_journal-610 29 3 ) ) PUNCT american_scientific_journal-610 29 4 = = SYM american_scientific_journal-610 30 1 𝑆. 𝑆. PROPN american_scientific_journal-610 30 2 as as ADP american_scientific_journal-610 30 3 a a DET american_scientific_journal-610 30 4 corollary corollary NOUN american_scientific_journal-610 30 5 from from ADP american_scientific_journal-610 30 6 this this DET american_scientific_journal-610 30 7 theorem theorem NOUN american_scientific_journal-610 30 8 we we PRON american_scientific_journal-610 30 9 mentioned mention VERB american_scientific_journal-610 30 10 that that SCONJ american_scientific_journal-610 30 11 the the DET american_scientific_journal-610 30 12 set set NOUN american_scientific_journal-610 30 13 of of ADP american_scientific_journal-610 30 14 rational rational ADJ american_scientific_journal-610 30 15 numbers number NOUN american_scientific_journal-610 30 16 is be AUX american_scientific_journal-610 30 17 a a DET american_scientific_journal-610 30 18 convergence convergence NOUN american_scientific_journal-610 30 19 set set NOUN american_scientific_journal-610 30 20 , , PUNCT american_scientific_journal-610 30 21 we we PRON american_scientific_journal-610 30 22 discuss discuss VERB american_scientific_journal-610 30 23 the the DET american_scientific_journal-610 30 24 proof proof NOUN american_scientific_journal-610 30 25 in in ADP american_scientific_journal-610 30 26 this this DET american_scientific_journal-610 30 27 paper paper NOUN american_scientific_journal-610 30 28 . . PUNCT american_scientific_journal-610 31 1 2.5 2.5 NUM american_scientific_journal-610 31 2 . . PUNCT american_scientific_journal-610 32 1 corollary corollary VERB american_scientific_journal-610 32 2 the the DET american_scientific_journal-610 32 3 set set VERB american_scientific_journal-610 32 4 β„š β„š PROPN american_scientific_journal-610 32 5 of of ADP american_scientific_journal-610 32 6 rational rational ADJ american_scientific_journal-610 32 7 numbers number NOUN american_scientific_journal-610 32 8 is be AUX american_scientific_journal-610 32 9 a a DET american_scientific_journal-610 32 10 convergence convergence NOUN american_scientific_journal-610 32 11 set set VERB american_scientific_journal-610 32 12 . . PUNCT american_scientific_journal-610 33 1 proof proof NOUN american_scientific_journal-610 33 2 . . PUNCT american_scientific_journal-610 34 1 πΉπ‘œπ‘Ÿ πΉπ‘œπ‘Ÿ NUM american_scientific_journal-610 34 2 𝑛 𝑛 PROPN american_scientific_journal-610 34 3 ∈ ∈ PROPN american_scientific_journal-610 34 4 β„• β„• PROPN american_scientific_journal-610 34 5 , , PUNCT american_scientific_journal-610 34 6 by by ADP american_scientific_journal-610 34 7 weierstrass weierstrass NOUN american_scientific_journal-610 34 8 theorem theorem NOUN american_scientific_journal-610 34 9 see see VERB american_scientific_journal-610 34 10 [ [ X american_scientific_journal-610 34 11 10 10 NUM american_scientific_journal-610 34 12 ] ] X american_scientific_journal-610 34 13 any any DET american_scientific_journal-610 34 14 function function NOUN american_scientific_journal-610 34 15 on on ADP american_scientific_journal-610 34 16 the the DET american_scientific_journal-610 34 17 closed closed ADJ american_scientific_journal-610 34 18 disc disc NOUN american_scientific_journal-610 34 19 { { PUNCT american_scientific_journal-610 34 20 𝑧 𝑧 PRON american_scientific_journal-610 34 21 ∈ ∈ NOUN american_scientific_journal-610 34 22 β„‚ β„‚ PROPN american_scientific_journal-610 34 23 ∢ ∢ NOUN american_scientific_journal-610 34 24 |𝑧| |𝑧| ADJ american_scientific_journal-610 34 25 ≀ ≀ NOUN american_scientific_journal-610 34 26 𝑛 𝑛 VERB american_scientific_journal-610 34 27 } } PUNCT american_scientific_journal-610 34 28 can can AUX american_scientific_journal-610 34 29 be be AUX american_scientific_journal-610 34 30 approximated approximate VERB american_scientific_journal-610 34 31 arbitrarily arbitrarily ADV american_scientific_journal-610 34 32 by by ADP american_scientific_journal-610 34 33 a a DET american_scientific_journal-610 34 34 polynomial polynomial NOUN american_scientific_journal-610 34 35 . . PUNCT american_scientific_journal-610 35 1 so so ADV american_scientific_journal-610 35 2 for for SCONJ american_scientific_journal-610 35 3 ( ( PUNCT american_scientific_journal-610 35 4 𝑛𝑛 𝑛𝑛 NOUN american_scientific_journal-610 35 5 sin sin PROPN american_scientific_journal-610 35 6 𝑛!πœ‹π‘§ 𝑛!πœ‹π‘§ VERB american_scientific_journal-610 35 7 ) ) PUNCT american_scientific_journal-610 35 8 , , PUNCT american_scientific_journal-610 35 9 n n PRON american_scientific_journal-610 35 10 πœ–β„• πœ–β„• NOUN american_scientific_journal-610 35 11 , , PUNCT american_scientific_journal-610 35 12 one one PRON american_scientific_journal-610 35 13 can can AUX american_scientific_journal-610 35 14 find find VERB american_scientific_journal-610 35 15 a a DET american_scientific_journal-610 35 16 polynomial polynomial ADJ american_scientific_journal-610 35 17 𝑃𝑛(z 𝑃𝑛(z NOUN american_scientific_journal-610 35 18 ) ) PUNCT american_scientific_journal-610 35 19 such such ADJ american_scientific_journal-610 35 20 that that SCONJ american_scientific_journal-610 35 21 |𝑷𝒏(𝒛 |𝑷𝒏(𝒛 PROPN american_scientific_journal-610 35 22 ) ) PUNCT american_scientific_journal-610 35 23 βˆ’ βˆ’ PROPN american_scientific_journal-610 35 24 𝒏𝒏 𝒏𝒏 NOUN american_scientific_journal-610 35 25 𝐬𝐒𝐧 𝐬𝐒𝐧 ADP american_scientific_journal-610 35 26 𝒏!𝝅𝒛)| 𝒏!𝝅𝒛)| PROPN american_scientific_journal-610 35 27 < < X american_scientific_journal-610 35 28 𝟏 𝟏 NUM american_scientific_journal-610 35 29 𝒏 𝒏 PROPN american_scientific_journal-610 35 30 , , PUNCT american_scientific_journal-610 35 31 for for ADP american_scientific_journal-610 35 32 |𝒛| |𝒛| PRON american_scientific_journal-610 35 33 ≀ ≀ NUM american_scientific_journal-610 35 34 𝒏. 𝒏. NOUN american_scientific_journal-610 35 35 170 170 NUM american_scientific_journal-610 35 36 american american PROPN american_scientific_journal-610 35 37 scientific scientific ADJ american_scientific_journal-610 35 38 research research NOUN american_scientific_journal-610 35 39 journal journal NOUN american_scientific_journal-610 35 40 for for ADP american_scientific_journal-610 35 41 engineering engineering NOUN american_scientific_journal-610 35 42 , , PUNCT american_scientific_journal-610 35 43 technology technology NOUN american_scientific_journal-610 35 44 , , PUNCT american_scientific_journal-610 35 45 and and CCONJ american_scientific_journal-610 35 46 sciences science NOUN american_scientific_journal-610 35 47 ( ( PUNCT american_scientific_journal-610 35 48 asrjets asrjet NOUN american_scientific_journal-610 35 49 ) ) PUNCT american_scientific_journal-610 35 50 ( ( PUNCT american_scientific_journal-610 35 51 2015 2015 NUM american_scientific_journal-610 35 52 ) ) PUNCT american_scientific_journal-610 35 53 volume volume NOUN american_scientific_journal-610 35 54 11 11 NUM american_scientific_journal-610 35 55 , , PUNCT american_scientific_journal-610 35 56 no no DET american_scientific_journal-610 35 57 1 1 NUM american_scientific_journal-610 35 58 , , PUNCT american_scientific_journal-610 35 59 pp pp ADV american_scientific_journal-610 35 60 169 169 NUM american_scientific_journal-610 35 61 - - SYM american_scientific_journal-610 35 62 172 172 NUM american_scientific_journal-610 35 63 let let VERB american_scientific_journal-610 35 64 the the DET american_scientific_journal-610 35 65 formal formal ADJ american_scientific_journal-610 35 66 series series NOUN american_scientific_journal-610 35 67 𝑭(𝒛 𝑭(𝒛 PROPN american_scientific_journal-610 35 68 , , PUNCT american_scientific_journal-610 35 69 𝒕 𝒕 X american_scientific_journal-610 35 70 ) ) PUNCT american_scientific_journal-610 35 71 = = PUNCT american_scientific_journal-610 36 1 βˆ‘ βˆ‘ PUNCT american_scientific_journal-610 36 2 𝑷𝒏(𝒛)π’•π’βˆž 𝑷𝒏(𝒛)π’•π’βˆž PUNCT american_scientific_journal-610 36 3 𝒏=𝟎 𝒏=𝟎 PUNCT american_scientific_journal-610 36 4 . . PUNCT american_scientific_journal-610 37 1 now now ADV american_scientific_journal-610 37 2 for for ADP american_scientific_journal-610 37 3 every every DET american_scientific_journal-610 37 4 𝑧 𝑧 DET american_scientific_journal-610 37 5 ∈ ∈ PROPN american_scientific_journal-610 37 6 β„‚ β„‚ PROPN american_scientific_journal-610 37 7 β„š,⁄ β„š,⁄ VERB american_scientific_journal-610 37 8 the the DET american_scientific_journal-610 37 9 bounded bounded ADJ american_scientific_journal-610 37 10 sequence sequence NOUN american_scientific_journal-610 37 11 { { PUNCT american_scientific_journal-610 37 12 sin𝑛!πœ‹π‘§)}𝑛=1 sin𝑛!πœ‹π‘§)}𝑛=1 NOUN american_scientific_journal-610 37 13 ∞ ∞ PROPN american_scientific_journal-610 37 14 is be AUX american_scientific_journal-610 37 15 divergent divergent ADJ american_scientific_journal-610 37 16 as as SCONJ american_scientific_journal-610 37 17 𝑛 𝑛 PROPN american_scientific_journal-610 37 18 extends extend VERB american_scientific_journal-610 37 19 to to PART american_scientific_journal-610 37 20 infinity infinity NOUN american_scientific_journal-610 37 21 . . PUNCT american_scientific_journal-610 38 1 suppose suppose VERB american_scientific_journal-610 38 2 there there PRON american_scientific_journal-610 38 3 exist exist VERB american_scientific_journal-610 38 4 a a DET american_scientific_journal-610 38 5 sub sub NOUN american_scientific_journal-610 38 6 - - NOUN american_scientific_journal-610 38 7 sequence sequence NOUN american_scientific_journal-610 38 8 of of ADP american_scientific_journal-610 38 9 οΏ½ οΏ½ PROPN american_scientific_journal-610 38 10 𝑛𝑗 𝑛𝑗 NOUN american_scientific_journal-610 38 11 οΏ½ οΏ½ NOUN american_scientific_journal-610 38 12 𝑗=1 𝑗=1 NOUN american_scientific_journal-610 38 13 ∞ ∞ NUM american_scientific_journal-610 38 14 such such ADJ american_scientific_journal-610 38 15 that that SCONJ american_scientific_journal-610 38 16 limπ‘—β†’βˆž limπ‘—β†’βˆž PROPN american_scientific_journal-610 38 17 sin sin NOUN american_scientific_journal-610 38 18 𝑛!πœ‹π‘§ 𝑛!πœ‹π‘§ VERB american_scientific_journal-610 38 19 = = SYM american_scientific_journal-610 38 20 πœ† πœ† NOUN american_scientific_journal-610 38 21 , , PUNCT american_scientific_journal-610 38 22 where where SCONJ american_scientific_journal-610 38 23 πœ† πœ† NOUN american_scientific_journal-610 38 24 is be AUX american_scientific_journal-610 38 25 a a DET american_scientific_journal-610 38 26 complex complex ADJ american_scientific_journal-610 38 27 number number NOUN american_scientific_journal-610 38 28 . . PUNCT american_scientific_journal-610 39 1 using use VERB american_scientific_journal-610 39 2 the the DET american_scientific_journal-610 39 3 previous previous ADJ american_scientific_journal-610 39 4 inequality inequality NOUN american_scientific_journal-610 39 5 we we PRON american_scientific_journal-610 39 6 get get VERB american_scientific_journal-610 39 7 |𝑷𝒏𝒋(𝒛 |𝑷𝒏𝒋(𝒛 ADV american_scientific_journal-610 39 8 ) ) PUNCT american_scientific_journal-610 40 1 βˆ’ βˆ’ PROPN american_scientific_journal-610 40 2 𝒏𝒋 𝒏𝒋 PROPN american_scientific_journal-610 40 3 𝒏𝒋 𝒏𝒋 PROPN american_scientific_journal-610 40 4 𝐬𝐒𝐧 𝐬𝐒𝐧 ADP american_scientific_journal-610 40 5 𝒏𝒋!𝝅𝒛)| 𝒏𝒋!𝝅𝒛)| PROPN american_scientific_journal-610 40 6 < < X american_scientific_journal-610 40 7 𝟏 𝟏 PROPN american_scientific_journal-610 40 8 𝒏𝒋 𝒏𝒋 NOUN american_scientific_journal-610 40 9 , , PUNCT american_scientific_journal-610 40 10 for for ADP american_scientific_journal-610 40 11 j j PROPN american_scientific_journal-610 40 12 ≀ ≀ PROPN american_scientific_journal-610 40 13 |𝒛| |𝒛| PROPN american_scientific_journal-610 40 14 , , PUNCT american_scientific_journal-610 40 15 which which PRON american_scientific_journal-610 40 16 gives give VERB american_scientific_journal-610 40 17 that that DET american_scientific_journal-610 40 18 |𝑷𝒏𝒋(𝒛)| |𝑷𝒏𝒋(𝒛)| NOUN american_scientific_journal-610 40 19 β‰₯ β‰₯ NOUN american_scientific_journal-610 40 20 |𝒏𝒋 |𝒏𝒋 X american_scientific_journal-610 40 21 𝒏𝒋 𝒏𝒋 PROPN american_scientific_journal-610 40 22 𝐬𝐒𝐧 𝐬𝐒𝐧 ADP american_scientific_journal-610 40 23 𝒏𝒋!𝝅𝒛| 𝒏𝒋!𝝅𝒛| PROPN american_scientific_journal-610 40 24 𝟏 𝟏 PROPN american_scientific_journal-610 40 25 𝒏𝒋 𝒏𝒋 NOUN american_scientific_journal-610 40 26 , , PUNCT american_scientific_journal-610 40 27 βˆ€ βˆ€ X american_scientific_journal-610 40 28 j j PROPN american_scientific_journal-610 40 29 β‰₯ β‰₯ NUM american_scientific_journal-610 40 30 |𝒛| |𝒛| PROPN american_scientific_journal-610 40 31 . . PUNCT american_scientific_journal-610 41 1 |𝑷𝒏𝒋(𝒛 |𝑷𝒏𝒋(𝒛 ADV american_scientific_journal-610 41 2 ) ) PUNCT american_scientific_journal-610 41 3 β‰₯ β‰₯ NOUN american_scientific_journal-610 41 4 𝟏 𝟏 NUM american_scientific_journal-610 41 5 𝟐 𝟐 NUM american_scientific_journal-610 41 6 |𝝀|𝒏𝒋 |𝝀|𝒏𝒋 PROPN american_scientific_journal-610 41 7 𝒏𝒋 𝒏𝒋 NOUN american_scientific_journal-610 41 8 , , PUNCT american_scientific_journal-610 41 9 for for ADP american_scientific_journal-610 41 10 𝑛𝑗 𝑛𝑗 ADP american_scientific_journal-610 41 11 large large ADJ american_scientific_journal-610 41 12 enough enough ADV american_scientific_journal-610 41 13 , , PUNCT american_scientific_journal-610 41 14 𝐹(𝑧 𝐹(𝑧 NUM american_scientific_journal-610 41 15 , , PUNCT american_scientific_journal-610 41 16 𝑑 𝑑 NOUN american_scientific_journal-610 41 17 ) ) PUNCT american_scientific_journal-610 42 1 is be AUX american_scientific_journal-610 42 2 divergent divergent ADJ american_scientific_journal-610 42 3 βˆ€π‘› βˆ€π‘› NOUN american_scientific_journal-610 42 4 ∈ ∈ PROPN american_scientific_journal-610 42 5 β„•. β„•. PROPN american_scientific_journal-610 42 6 on on ADP american_scientific_journal-610 42 7 the the DET american_scientific_journal-610 42 8 other other ADJ american_scientific_journal-610 42 9 hand hand NOUN american_scientific_journal-610 42 10 , , PUNCT american_scientific_journal-610 42 11 for for SCONJ american_scientific_journal-610 42 12 x∈ x∈ PROPN american_scientific_journal-610 42 13 β„š β„š PROPN american_scientific_journal-610 42 14 let let VERB american_scientific_journal-610 42 15 x x X american_scientific_journal-610 42 16 = = SYM american_scientific_journal-610 43 1 𝑙 𝑙 PROPN american_scientific_journal-610 43 2 π‘š π‘š X american_scientific_journal-610 43 3 where where SCONJ american_scientific_journal-610 43 4 𝑙,π‘š 𝑙,π‘š PROPN american_scientific_journal-610 43 5 ∈ ∈ PROPN american_scientific_journal-610 43 6 β„€ β„€ PROPN american_scientific_journal-610 43 7 and and CCONJ american_scientific_journal-610 43 8 the the DET american_scientific_journal-610 43 9 gcd(𝑙,π‘š gcd(𝑙,π‘š NOUN american_scientific_journal-610 43 10 ) ) PUNCT american_scientific_journal-610 43 11 = = SYM american_scientific_journal-610 44 1 1 1 X american_scientific_journal-610 44 2 . . PUNCT american_scientific_journal-610 44 3 then then ADV american_scientific_journal-610 44 4 for for ADP american_scientific_journal-610 44 5 𝑛 𝑛 PROPN american_scientific_journal-610 44 6 > > PUNCT american_scientific_journal-610 44 7 π‘š π‘š X american_scientific_journal-610 44 8 |𝑷𝒏(𝒙)| |𝑷𝒏(𝒙)| NOUN american_scientific_journal-610 44 9 ≀ ≀ PUNCT american_scientific_journal-610 44 10 | | ADV american_scientific_journal-610 44 11 𝒏𝒏 𝒏𝒏 NOUN american_scientific_journal-610 44 12 𝐬𝐒𝐧 𝐬𝐒𝐧 ADP american_scientific_journal-610 44 13 𝒏!𝝅𝒙 𝒏!𝝅𝒙 NOUN american_scientific_journal-610 44 14 ) ) PUNCT american_scientific_journal-610 44 15 + + CCONJ american_scientific_journal-610 44 16 𝟏 𝟏 NUM american_scientific_journal-610 44 17 𝒏 𝒏 VERB american_scientific_journal-610 44 18 | | NOUN american_scientific_journal-610 44 19 = = SYM american_scientific_journal-610 44 20 𝟎 𝟎 PROPN american_scientific_journal-610 44 21 + + CCONJ american_scientific_journal-610 44 22 𝟏 𝟏 NUM american_scientific_journal-610 44 23 𝒏 𝒏 PROPN american_scientific_journal-610 44 24 . . PUNCT american_scientific_journal-610 45 1 so so ADV american_scientific_journal-610 45 2 𝐹(π‘₯ 𝐹(π‘₯ NUM american_scientific_journal-610 45 3 , , PUNCT american_scientific_journal-610 45 4 𝑑 𝑑 NOUN american_scientific_journal-610 45 5 ) ) PUNCT american_scientific_journal-610 45 6 is be AUX american_scientific_journal-610 45 7 convergence convergence NOUN american_scientific_journal-610 45 8 . . PUNCT american_scientific_journal-610 46 1 the the DET american_scientific_journal-610 46 2 corollary corollary NOUN american_scientific_journal-610 46 3 indicates indicate VERB american_scientific_journal-610 46 4 that that SCONJ american_scientific_journal-610 46 5 the the DET american_scientific_journal-610 46 6 whole whole ADJ american_scientific_journal-610 46 7 rational rational ADJ american_scientific_journal-610 46 8 numbers number NOUN american_scientific_journal-610 46 9 β„š β„š PROPN american_scientific_journal-610 46 10 is be AUX american_scientific_journal-610 46 11 a a DET american_scientific_journal-610 46 12 convergence convergence NOUN american_scientific_journal-610 46 13 set set NOUN american_scientific_journal-610 46 14 . . PUNCT american_scientific_journal-610 47 1 now now ADV american_scientific_journal-610 47 2 since since SCONJ american_scientific_journal-610 47 3 the the DET american_scientific_journal-610 47 4 unit unit NOUN american_scientific_journal-610 47 5 interval interval NOUN american_scientific_journal-610 47 6 [ [ X american_scientific_journal-610 47 7 0 0 NUM american_scientific_journal-610 47 8 , , PUNCT american_scientific_journal-610 47 9 1 1 NUM american_scientific_journal-610 47 10 ] ] PUNCT american_scientific_journal-610 47 11 is be AUX american_scientific_journal-610 47 12 compact compact ADJ american_scientific_journal-610 47 13 and and CCONJ american_scientific_journal-610 47 14 simply simply ADV american_scientific_journal-610 47 15 connected connect VERB american_scientific_journal-610 47 16 set set VERB american_scientific_journal-610 47 17 it it PRON american_scientific_journal-610 47 18 's be AUX american_scientific_journal-610 47 19 a a DET american_scientific_journal-610 47 20 convergence convergence NOUN american_scientific_journal-610 47 21 set set VERB american_scientific_journal-610 47 22 . . PUNCT american_scientific_journal-610 48 1 in in ADP american_scientific_journal-610 48 2 [ [ X american_scientific_journal-610 48 3 3 3 X american_scientific_journal-610 48 4 ] ] PUNCT american_scientific_journal-610 48 5 we we PRON american_scientific_journal-610 48 6 proved prove VERB american_scientific_journal-610 48 7 that that SCONJ american_scientific_journal-610 48 8 a a DET american_scientific_journal-610 48 9 finite finite ADJ american_scientific_journal-610 48 10 intersection intersection NOUN american_scientific_journal-610 48 11 of of ADP american_scientific_journal-610 48 12 convergence convergence NOUN american_scientific_journal-610 48 13 sets set NOUN american_scientific_journal-610 48 14 is be AUX american_scientific_journal-610 48 15 a a DET american_scientific_journal-610 48 16 convergence convergence NOUN american_scientific_journal-610 48 17 set set NOUN american_scientific_journal-610 48 18 , , PUNCT american_scientific_journal-610 48 19 so so CCONJ american_scientific_journal-610 48 20 the the DET american_scientific_journal-610 48 21 intersection intersection NOUN american_scientific_journal-610 48 22 of of ADP american_scientific_journal-610 48 23 the the DET american_scientific_journal-610 48 24 β„š β„š PROPN american_scientific_journal-610 48 25 and and CCONJ american_scientific_journal-610 48 26 the the DET american_scientific_journal-610 48 27 unit unit NOUN american_scientific_journal-610 48 28 interval interval NOUN american_scientific_journal-610 48 29 is be AUX american_scientific_journal-610 48 30 a a DET american_scientific_journal-610 48 31 convergence convergence NOUN american_scientific_journal-610 48 32 set set VERB american_scientific_journal-610 48 33 . . PUNCT american_scientific_journal-610 49 1 2.6 2.6 NUM american_scientific_journal-610 49 2 . . PUNCT american_scientific_journal-610 49 3 example example NOUN american_scientific_journal-610 49 4 the the DET american_scientific_journal-610 49 5 set set NOUN american_scientific_journal-610 49 6 of of ADP american_scientific_journal-610 49 7 rational rational ADJ american_scientific_journal-610 49 8 number number NOUN american_scientific_journal-610 49 9 in in ADP american_scientific_journal-610 49 10 the the DET american_scientific_journal-610 49 11 unit unit NOUN american_scientific_journal-610 49 12 interval interval NOUN american_scientific_journal-610 49 13 , , PUNCT american_scientific_journal-610 49 14 k k PROPN american_scientific_journal-610 50 1 = = PUNCT american_scientific_journal-610 51 1 [ [ X american_scientific_journal-610 51 2 0 0 NUM american_scientific_journal-610 51 3 , , PUNCT american_scientific_journal-610 51 4 1 1 NUM american_scientific_journal-610 51 5 ] ] PUNCT american_scientific_journal-610 51 6 \ \ PROPN american_scientific_journal-610 51 7 β„š β„š PROPN american_scientific_journal-610 51 8 , , PUNCT american_scientific_journal-610 51 9 is be AUX american_scientific_journal-610 51 10 a a DET american_scientific_journal-610 51 11 convergence convergence NOUN american_scientific_journal-610 51 12 set set VERB american_scientific_journal-610 51 13 . . PUNCT american_scientific_journal-610 52 1 3 3 X american_scientific_journal-610 52 2 . . X american_scientific_journal-610 52 3 conclusion conclusion NOUN american_scientific_journal-610 52 4 in in ADP american_scientific_journal-610 52 5 this this DET american_scientific_journal-610 52 6 paper paper NOUN american_scientific_journal-610 52 7 we we PRON american_scientific_journal-610 52 8 find find VERB american_scientific_journal-610 52 9 that that SCONJ american_scientific_journal-610 52 10 β„š β„š PROPN american_scientific_journal-610 52 11 is be AUX american_scientific_journal-610 52 12 a a DET american_scientific_journal-610 52 13 convergence convergence NOUN american_scientific_journal-610 52 14 set set NOUN american_scientific_journal-610 52 15 , , PUNCT american_scientific_journal-610 52 16 which which PRON american_scientific_journal-610 52 17 implies imply VERB american_scientific_journal-610 52 18 that that SCONJ american_scientific_journal-610 52 19 β„š β„š PROPN american_scientific_journal-610 52 20 is be AUX american_scientific_journal-610 52 21 a a DET american_scientific_journal-610 52 22 quasi quasi ADJ american_scientific_journal-610 52 23 - - ADJ american_scientific_journal-610 52 24 simply simply ADV american_scientific_journal-610 52 25 - - PUNCT american_scientific_journal-610 52 26 connected connect VERB american_scientific_journal-610 52 27 set set NOUN american_scientific_journal-610 52 28 , , PUNCT american_scientific_journal-610 52 29 looking look VERB american_scientific_journal-610 52 30 for for ADP american_scientific_journal-610 52 31 new new ADJ american_scientific_journal-610 52 32 sets set NOUN american_scientific_journal-610 52 33 which which PRON american_scientific_journal-610 52 34 is be AUX american_scientific_journal-610 52 35 convergence convergence NOUN american_scientific_journal-610 52 36 and and CCONJ american_scientific_journal-610 52 37 have have VERB american_scientific_journal-610 52 38 other other ADJ american_scientific_journal-610 52 39 formal formal ADJ american_scientific_journal-610 52 40 power power NOUN american_scientific_journal-610 52 41 series series NOUN american_scientific_journal-610 52 42 properties property NOUN american_scientific_journal-610 52 43 . . PUNCT american_scientific_journal-610 53 1 171 171 NUM american_scientific_journal-610 53 2 american american ADJ american_scientific_journal-610 53 3 scientific scientific ADJ american_scientific_journal-610 53 4 research research NOUN american_scientific_journal-610 53 5 journal journal NOUN american_scientific_journal-610 53 6 for for ADP american_scientific_journal-610 53 7 engineering engineering NOUN american_scientific_journal-610 53 8 , , PUNCT american_scientific_journal-610 53 9 technology technology NOUN american_scientific_journal-610 53 10 , , PUNCT american_scientific_journal-610 53 11 and and CCONJ american_scientific_journal-610 53 12 sciences science NOUN american_scientific_journal-610 53 13 ( ( PUNCT american_scientific_journal-610 53 14 asrjets asrjet NOUN american_scientific_journal-610 53 15 ) ) PUNCT american_scientific_journal-610 53 16 ( ( PUNCT american_scientific_journal-610 53 17 2015 2015 NUM american_scientific_journal-610 53 18 ) ) PUNCT american_scientific_journal-610 53 19 volume volume NOUN american_scientific_journal-610 53 20 11 11 NUM american_scientific_journal-610 53 21 , , PUNCT american_scientific_journal-610 53 22 no no DET american_scientific_journal-610 53 23 1 1 NUM american_scientific_journal-610 53 24 , , PUNCT american_scientific_journal-610 53 25 pp pp ADV american_scientific_journal-610 53 26 169 169 NUM american_scientific_journal-610 53 27 - - SYM american_scientific_journal-610 53 28 172 172 NUM american_scientific_journal-610 53 29 references reference NOUN american_scientific_journal-610 53 30 [ [ X american_scientific_journal-610 53 31 1 1 NUM american_scientific_journal-610 53 32 ] ] X american_scientific_journal-610 53 33 s.s s.s PROPN american_scientific_journal-610 53 34 . . PROPN american_scientific_journal-610 53 35 abhyankar abhyankar PROPN american_scientific_journal-610 53 36 , , PUNCT american_scientific_journal-610 53 37 t.t t.t PROPN american_scientific_journal-610 53 38 . . PROPN american_scientific_journal-610 53 39 moh moh PROPN american_scientific_journal-610 53 40 , , PUNCT american_scientific_journal-610 53 41 '' '' PUNCT american_scientific_journal-610 53 42 a a DET american_scientific_journal-610 53 43 reduction reduction NOUN american_scientific_journal-610 53 44 theorem theorem VERB american_scientific_journal-610 53 45 for for ADP american_scientific_journal-610 53 46 divergent divergent ADJ american_scientific_journal-610 53 47 power power NOUN american_scientific_journal-610 53 48 series series NOUN american_scientific_journal-610 53 49 '' '' PUNCT american_scientific_journal-610 53 50 , , PUNCT american_scientific_journal-610 53 51 j. j. PROPN american_scientific_journal-610 53 52 reine reine PROPN american_scientific_journal-610 53 53 angew angew PROPN american_scientific_journal-610 53 54 . . PUNCT american_scientific_journal-610 54 1 math math NOUN american_scientific_journal-610 54 2 . . PUNCT american_scientific_journal-610 54 3 , , PUNCT american_scientific_journal-610 54 4 241(1970 241(1970 NUM american_scientific_journal-610 54 5 ) ) PUNCT american_scientific_journal-610 54 6 , , PUNCT american_scientific_journal-610 54 7 pp pp ADP american_scientific_journal-610 54 8 27 27 NUM american_scientific_journal-610 54 9 - - SYM american_scientific_journal-610 54 10 33 33 NUM american_scientific_journal-610 54 11 . . PUNCT american_scientific_journal-610 55 1 [ [ X american_scientific_journal-610 55 2 2 2 NUM american_scientific_journal-610 55 3 ] ] X american_scientific_journal-610 55 4 b.l b.l PROPN american_scientific_journal-610 55 5 . . PROPN american_scientific_journal-610 55 6 fridman fridman PROPN american_scientific_journal-610 55 7 , , PUNCT american_scientific_journal-610 55 8 d. d. PROPN american_scientific_journal-610 55 9 ma ma PROPN american_scientific_journal-610 55 10 , , PUNCT american_scientific_journal-610 55 11 '' '' PUNCT american_scientific_journal-610 55 12 osgood osgood NOUN american_scientific_journal-610 55 13 - - PUNCT american_scientific_journal-610 55 14 hartogs hartog NOUN american_scientific_journal-610 55 15 type type NOUN american_scientific_journal-610 55 16 properties property NOUN american_scientific_journal-610 55 17 of of ADP american_scientific_journal-610 55 18 power power NOUN american_scientific_journal-610 55 19 series series NOUN american_scientific_journal-610 55 20 and and CCONJ american_scientific_journal-610 55 21 smooth smooth ADJ american_scientific_journal-610 55 22 functions function NOUN american_scientific_journal-610 55 23 '' '' PUNCT american_scientific_journal-610 55 24 , , PUNCT american_scientific_journal-610 55 25 pacific pacific PROPN american_scientific_journal-610 55 26 j. j. PROPN american_scientific_journal-610 55 27 math math PROPN american_scientific_journal-610 55 28 . . PROPN american_scientific_journal-610 55 29 , , PUNCT american_scientific_journal-610 55 30 251(2011 251(2011 NUM american_scientific_journal-610 55 31 ) ) PUNCT american_scientific_journal-610 55 32 , , PUNCT american_scientific_journal-610 55 33 pp pp ADV american_scientific_journal-610 55 34 67 67 NUM american_scientific_journal-610 55 35 - - SYM american_scientific_journal-610 55 36 79 79 NUM american_scientific_journal-610 55 37 . . PUNCT american_scientific_journal-610 56 1 [ [ X american_scientific_journal-610 56 2 3 3 X american_scientific_journal-610 56 3 ] ] X american_scientific_journal-610 56 4 b. b. PROPN american_scientific_journal-610 56 5 al al PROPN american_scientific_journal-610 56 6 - - PUNCT american_scientific_journal-610 56 7 shutnawi shutnawi PROPN american_scientific_journal-610 56 8 , , PUNCT american_scientific_journal-610 56 9 '' '' PUNCT american_scientific_journal-610 56 10 on on ADP american_scientific_journal-610 56 11 convergence convergence NOUN american_scientific_journal-610 56 12 sets set NOUN american_scientific_journal-610 56 13 of of ADP american_scientific_journal-610 56 14 formal formal ADJ american_scientific_journal-610 56 15 power power NOUN american_scientific_journal-610 56 16 series series NOUN american_scientific_journal-610 56 17 '' '' PUNCT american_scientific_journal-610 56 18 , , PUNCT american_scientific_journal-610 56 19 phd phd NOUN american_scientific_journal-610 56 20 dissertation dissertation NOUN american_scientific_journal-610 56 21 , , PUNCT american_scientific_journal-610 56 22 usa usa PROPN american_scientific_journal-610 56 23 , , PUNCT american_scientific_journal-610 56 24 2013 2013 NUM american_scientific_journal-610 57 1 [ [ X american_scientific_journal-610 57 2 4 4 NUM american_scientific_journal-610 57 3 ] ] X american_scientific_journal-610 57 4 b.l b.l PROPN american_scientific_journal-610 57 5 . . PROPN american_scientific_journal-610 57 6 fridman fridman PROPN american_scientific_journal-610 57 7 , , PUNCT american_scientific_journal-610 57 8 d. d. PROPN american_scientific_journal-610 57 9 ma ma PROPN american_scientific_journal-610 57 10 , , PUNCT american_scientific_journal-610 57 11 t.s t.s PROPN american_scientific_journal-610 57 12 . . PROPN american_scientific_journal-610 57 13 neelon neelon PROPN american_scientific_journal-610 57 14 , , PUNCT american_scientific_journal-610 57 15 '' '' PUNCT american_scientific_journal-610 57 16 nonlinear nonlinear ADJ american_scientific_journal-610 57 17 convergence convergence NOUN american_scientific_journal-610 57 18 sets set NOUN american_scientific_journal-610 57 19 of of ADP american_scientific_journal-610 57 20 divergent divergent ADJ american_scientific_journal-610 57 21 power power NOUN american_scientific_journal-610 57 22 series series NOUN american_scientific_journal-610 57 23 '' '' PUNCT american_scientific_journal-610 57 24 , , PUNCT american_scientific_journal-610 57 25 preprint preprint NOUN american_scientific_journal-610 57 26 . . PUNCT american_scientific_journal-610 58 1 [ [ X american_scientific_journal-610 58 2 5 5 NUM american_scientific_journal-610 58 3 ] ] X american_scientific_journal-610 58 4 r.c r.c PROPN american_scientific_journal-610 58 5 . . PROPN american_scientific_journal-610 58 6 gunning gunning PROPN american_scientific_journal-610 58 7 , , PUNCT american_scientific_journal-610 58 8 introduction introduction NOUN american_scientific_journal-610 58 9 to to ADP american_scientific_journal-610 58 10 holomorphic holomorphic ADJ american_scientific_journal-610 58 11 functions function NOUN american_scientific_journal-610 58 12 of of ADP american_scientific_journal-610 58 13 several several ADJ american_scientific_journal-610 58 14 variables variable NOUN american_scientific_journal-610 58 15 , , PUNCT american_scientific_journal-610 58 16 2nd 2nd ADJ american_scientific_journal-610 58 17 ed ed NOUN american_scientific_journal-610 58 18 , , PUNCT american_scientific_journal-610 58 19 wadsworth wadsworth NOUN american_scientific_journal-610 58 20 and and CCONJ american_scientific_journal-610 58 21 brooks brooks PROPN american_scientific_journal-610 58 22 / / SYM american_scientific_journal-610 58 23 cole cole PROPN american_scientific_journal-610 58 24 , , PUNCT american_scientific_journal-610 58 25 belmont belmont PROPN american_scientific_journal-610 58 26 , , PUNCT american_scientific_journal-610 58 27 california california PROPN american_scientific_journal-610 58 28 , , PUNCT american_scientific_journal-610 58 29 1990 1990 NUM american_scientific_journal-610 58 30 . . PUNCT american_scientific_journal-610 59 1 pp10 pp10 PROPN american_scientific_journal-610 59 2 - - PUNCT american_scientific_journal-610 59 3 100 100 NUM american_scientific_journal-610 59 4 . . PUNCT american_scientific_journal-610 60 1 [ [ X american_scientific_journal-610 60 2 6 6 NUM american_scientific_journal-610 60 3 ] ] PUNCT american_scientific_journal-610 60 4 a. a. NOUN american_scientific_journal-610 60 5 klebano klebano PROPN american_scientific_journal-610 61 1 _ _ PROPN american_scientific_journal-610 61 2 , , PUNCT american_scientific_journal-610 61 3 in in ADP american_scientific_journal-610 61 4 the the DET american_scientific_journal-610 61 5 mandelbrot mandelbrot NOUN american_scientific_journal-610 61 6 set set NOUN american_scientific_journal-610 61 7 . . PUNCT american_scientific_journal-610 62 1 internet internet NOUN american_scientific_journal-610 62 2 : : PUNCT american_scientific_journal-610 62 3 http://home.comcast.net/_davejanelle http://home.comcast.net/_davejanelle PROPN american_scientific_journal-610 62 4 /mandel.pdf(2013 /mandel.pdf(2013 PROPN american_scientific_journal-610 62 5 ) ) PUNCT american_scientific_journal-610 62 6 . . PUNCT american_scientific_journal-610 63 1 [ [ X american_scientific_journal-610 63 2 7 7 X american_scientific_journal-610 63 3 ] ] X american_scientific_journal-610 63 4 p. p. PROPN american_scientific_journal-610 63 5 lelong lelong PROPN american_scientific_journal-610 63 6 , , PUNCT american_scientific_journal-610 63 7 '' '' PUNCT american_scientific_journal-610 63 8 on on ADP american_scientific_journal-610 63 9 a a DET american_scientific_journal-610 63 10 problem problem NOUN american_scientific_journal-610 63 11 of of ADP american_scientific_journal-610 63 12 m.a m.a PROPN american_scientific_journal-610 63 13 . . PROPN american_scientific_journal-610 63 14 zorn zorn PROPN american_scientific_journal-610 63 15 '' '' PUNCT american_scientific_journal-610 63 16 , , PUNCT american_scientific_journal-610 63 17 proc proc PROPN american_scientific_journal-610 63 18 . . PUNCT american_scientific_journal-610 63 19 amer amer PROPN american_scientific_journal-610 63 20 . . PUNCT american_scientific_journal-610 63 21 math math PROPN american_scientific_journal-610 63 22 . . PUNCT american_scientific_journal-610 64 1 soc soc PROPN american_scientific_journal-610 64 2 . . PUNCT american_scientific_journal-610 64 3 , , PUNCT american_scientific_journal-610 64 4 2(1951 2(1951 NUM american_scientific_journal-610 64 5 ) ) PUNCT american_scientific_journal-610 64 6 , , PUNCT american_scientific_journal-610 64 7 pp pp ADP american_scientific_journal-610 64 8 11 11 NUM american_scientific_journal-610 64 9 - - SYM american_scientific_journal-610 64 10 19 19 NUM american_scientific_journal-610 64 11 . . PUNCT american_scientific_journal-610 65 1 [ [ X american_scientific_journal-610 65 2 8 8 NUM american_scientific_journal-610 65 3 ] ] X american_scientific_journal-610 65 4 n. n. PROPN american_scientific_journal-610 65 5 levenberg levenberg PROPN american_scientific_journal-610 65 6 , , PUNCT american_scientific_journal-610 65 7 r.e r.e PROPN american_scientific_journal-610 65 8 . . PROPN american_scientific_journal-610 65 9 molzon molzon PROPN american_scientific_journal-610 65 10 , , PUNCT american_scientific_journal-610 65 11 '' '' PUNCT american_scientific_journal-610 65 12 convergence convergence NOUN american_scientific_journal-610 65 13 sets set NOUN american_scientific_journal-610 65 14 of of ADP american_scientific_journal-610 65 15 a a DET american_scientific_journal-610 65 16 formal formal ADJ american_scientific_journal-610 65 17 power power NOUN american_scientific_journal-610 65 18 series series NOUN american_scientific_journal-610 65 19 '' '' PUNCT american_scientific_journal-610 65 20 , , PUNCT american_scientific_journal-610 65 21 math math NOUN american_scientific_journal-610 65 22 . . PUNCT american_scientific_journal-610 66 1 z. z. PROPN american_scientific_journal-610 66 2 , , PUNCT american_scientific_journal-610 66 3 197(1988 197(1988 NUM american_scientific_journal-610 66 4 ) ) PUNCT american_scientific_journal-610 66 5 , , PUNCT american_scientific_journal-610 66 6 pp pp ADV american_scientific_journal-610 66 7 411 411 NUM american_scientific_journal-610 66 8 - - SYM american_scientific_journal-610 66 9 420 420 NUM american_scientific_journal-610 66 10 . . PUNCT american_scientific_journal-610 67 1 [ [ X american_scientific_journal-610 67 2 9 9 NUM american_scientific_journal-610 67 3 ] ] X american_scientific_journal-610 67 4 d. d. PROPN american_scientific_journal-610 67 5 ma ma PROPN american_scientific_journal-610 67 6 , , PUNCT american_scientific_journal-610 67 7 t.s t.s PROPN american_scientific_journal-610 67 8 . . PROPN american_scientific_journal-610 67 9 neelon neelon PROPN american_scientific_journal-610 67 10 , , PUNCT american_scientific_journal-610 67 11 '' '' PUNCT american_scientific_journal-610 67 12 on on ADP american_scientific_journal-610 67 13 convergence convergence NOUN american_scientific_journal-610 67 14 sets set NOUN american_scientific_journal-610 67 15 of of ADP american_scientific_journal-610 67 16 formal formal ADJ american_scientific_journal-610 67 17 power power NOUN american_scientific_journal-610 67 18 series series NOUN american_scientific_journal-610 67 19 '' '' PUNCT american_scientific_journal-610 67 20 , , PUNCT american_scientific_journal-610 67 21 preprint preprint NOUN american_scientific_journal-610 67 22 . . PUNCT american_scientific_journal-610 68 1 [ [ X american_scientific_journal-610 68 2 10 10 NUM american_scientific_journal-610 68 3 ] ] X american_scientific_journal-610 68 4 r.m r.m PROPN american_scientific_journal-610 68 5 . . PROPN american_scientific_journal-610 68 6 range range PROPN american_scientific_journal-610 68 7 , , PUNCT american_scientific_journal-610 68 8 holomorphic holomorphic ADJ american_scientific_journal-610 68 9 functions function NOUN american_scientific_journal-610 68 10 and and CCONJ american_scientific_journal-610 68 11 integral integral ADJ american_scientific_journal-610 68 12 representations representation NOUN american_scientific_journal-610 68 13 in in ADP american_scientific_journal-610 68 14 several several ADJ american_scientific_journal-610 68 15 complex complex ADJ american_scientific_journal-610 68 16 variables variable NOUN american_scientific_journal-610 68 17 , , PUNCT american_scientific_journal-610 68 18 2nd 2nd ADJ american_scientific_journal-610 68 19 ed ed NOUN american_scientific_journal-610 68 20 , , PUNCT american_scientific_journal-610 68 21 springer springer NOUN american_scientific_journal-610 68 22 , , PUNCT american_scientific_journal-610 68 23 new new PROPN american_scientific_journal-610 68 24 york york PROPN american_scientific_journal-610 68 25 , , PUNCT american_scientific_journal-610 68 26 1986 1986 NUM american_scientific_journal-610 68 27 . . PUNCT american_scientific_journal-610 69 1 pp pp ADP american_scientific_journal-610 69 2 35 35 NUM american_scientific_journal-610 69 3 - - SYM american_scientific_journal-610 69 4 39 39 NUM american_scientific_journal-610 69 5 . . PUNCT american_scientific_journal-610 70 1 [ [ X american_scientific_journal-610 70 2 11 11 NUM american_scientific_journal-610 70 3 ] ] PUNCT american_scientific_journal-610 70 4 a. a. NOUN american_scientific_journal-610 70 5 sathaye sathaye NOUN american_scientific_journal-610 70 6 , , PUNCT american_scientific_journal-610 70 7 '' '' PUNCT american_scientific_journal-610 70 8 convergence convergence NOUN american_scientific_journal-610 70 9 sets set NOUN american_scientific_journal-610 70 10 of of ADP american_scientific_journal-610 70 11 divergent divergent ADJ american_scientific_journal-610 70 12 power power NOUN american_scientific_journal-610 70 13 series series NOUN american_scientific_journal-610 70 14 '' '' PUNCT american_scientific_journal-610 70 15 , , PUNCT american_scientific_journal-610 70 16 j. j. PROPN american_scientific_journal-610 70 17 reine reine PROPN american_scientific_journal-610 70 18 angew angew PROPN american_scientific_journal-610 70 19 . . PUNCT american_scientific_journal-610 71 1 math math NOUN american_scientific_journal-610 71 2 . . PUNCT american_scientific_journal-610 71 3 , , PUNCT american_scientific_journal-610 71 4 283(1976 283(1976 NUM american_scientific_journal-610 71 5 ) ) PUNCT american_scientific_journal-610 71 6 , , PUNCT american_scientific_journal-610 71 7 86|98.kokyoroku 86|98.kokyoroku PROPN american_scientific_journal-610 71 8 in in ADP american_scientific_journal-610 71 9 math math NOUN american_scientific_journal-610 71 10 . . PUNCT american_scientific_journal-610 72 1 14 14 NUM american_scientific_journal-610 72 2 , , PUNCT american_scientific_journal-610 72 3 tokyo tokyo PROPN american_scientific_journal-610 72 4 , , PUNCT american_scientific_journal-610 72 5 1982 1982 NUM american_scientific_journal-610 72 6 . . PUNCT american_scientific_journal-610 73 1 [ [ X american_scientific_journal-610 73 2 12 12 NUM american_scientific_journal-610 73 3 ] ] PUNCT american_scientific_journal-610 73 4 k. k. PROPN american_scientific_journal-610 73 5 spallek spallek PROPN american_scientific_journal-610 73 6 , , PUNCT american_scientific_journal-610 73 7 p. p. NOUN american_scientific_journal-610 73 8 tworzewski tworzewski NOUN american_scientific_journal-610 73 9 , , PUNCT american_scientific_journal-610 73 10 t. t. PROPN american_scientific_journal-610 73 11 winiarski winiarski PROPN american_scientific_journal-610 73 12 , , PUNCT american_scientific_journal-610 73 13 '' '' PUNCT american_scientific_journal-610 73 14 osgood osgood PROPN american_scientific_journal-610 73 15 - - PUNCT american_scientific_journal-610 73 16 hartogs hartog NOUN american_scientific_journal-610 73 17 - - PUNCT american_scientific_journal-610 73 18 theorem theorem NOUN american_scientific_journal-610 73 19 of of ADP american_scientific_journal-610 73 20 mixed mixed ADJ american_scientific_journal-610 73 21 type type NOUN american_scientific_journal-610 73 22 '' '' PUNCT american_scientific_journal-610 73 23 , , PUNCT american_scientific_journal-610 73 24 math math NOUN american_scientific_journal-610 73 25 . . PUNCT american_scientific_journal-610 74 1 ann,288(1990 ann,288(1990 PROPN american_scientific_journal-610 74 2 ) ) PUNCT american_scientific_journal-610 74 3 , , PUNCT american_scientific_journal-610 74 4 pp pp ADP american_scientific_journal-610 74 5 75 75 NUM american_scientific_journal-610 74 6 - - SYM american_scientific_journal-610 74 7 88 88 NUM american_scientific_journal-610 74 8 . . PUNCT american_scientific_journal-610 75 1 172 172 NUM