American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ 169 β„š is a Convergence Set Basma Al-Shutnawia*, Mohammad Zannonb a Department of Mathematics, Tafila technical university, P.O. Box 179 , Tafila 66110, Jordan. b Department of Mathematics, Tafila technical university, P.O. Box179, Tafila 66110, Jordan. a basma@ttu.edu.jo bzanno1ms@gmail.com Abstract In this paper we consider the convergence sets of formal power series of the form 𝑓(𝑧, 𝑑) = βˆ‘ 𝑓𝑗(𝑧)π‘‘π‘—βˆž 𝑗=0 , where 𝑓𝑗(𝑧) are polynomials functions on a domain Ξ© in β„‚. A subset 𝐸 of Ξ© is said to be convergence set if there is a series 𝑓(𝑧, 𝑑) such that 𝐸 is exactly the set of points 𝑧 for which 𝑓(𝑧, 𝑑) converges as a power series in t in some neighborhood of the origin. We prove that β„š is a convergence set. Keywords: formal power series; convergence sets, β„š the set of rational numbers, quasi-simply-connected sets. 1. Introduction Let β„‚[z] be the set of polynomials in z, β„‚[[𝑧]] be the set of formal power series, and β„‚[𝑧][[𝑑]] be the set of formal power series in 𝑑 with coefficients being polynomials in 𝑧. In my dissertation [3] I considered the formal power series 𝒇(𝒛) = π’‚πŸŽ + βˆ‘ π’‚πœΆπŸπ’› 𝜢𝟏|𝜢𝟏|=𝟏 + βˆ‘ π’‚πœΆπŸπ’› 𝜢𝟐|𝜢𝟐|=𝟐 + β‹―+ βˆ‘ π’‚πœΆπ’π’› πœΆπ’|πœΆπ’|=𝒏 , Where 𝛼𝑗= (𝛼𝑗1,…, 𝛼𝑗𝑛) are the n-multiple index and |𝛼𝑗 | = |𝛼𝑗1| + β‹―+ �𝛼𝑗𝑛�. Many research concerning convergence (or formal) series [7, 1, 11, 8, 5]. The general description of these problems is given in [12]. Recently there were new researches concerning the power series when the coefficients are polynomials of two or more complex variables [2, 4, 9, 6]. Consider 𝐹(𝑧, 𝑑) = βˆ‘ π‘ƒπ‘š(𝑧)π‘‘π‘šβˆž π‘š=0 . Suppose that 𝐹(𝑧, 𝑑) as a power series in 𝑑 converges for 𝑧 in a set 𝐸 βŠ‚ ℂ𝑛. I mentioned in my dissertation that the set β„š of rational numbers is a convergence set in this paper we prove this corollary. ------------------------------------------------------------------------ * Corresponding author. E-mail address: basma@ttu.edu.jo. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 11, No 1, pp 169-172 2. β„š is a Convergence Sets 2.1. Definition [3] A power series 𝑓 ∈ β„‚[[𝑧1, … , 𝑧𝑛]] is said to be convergent if there is a constant C such that |π‘Žπ‘˜1,…,π‘˜π‘›| ≀ πΆπ‘˜1+β‹―+π‘˜π‘› for all (π‘˜1, … , π‘˜π‘›) β‰  (0, … ,0). A power series 𝑓 is said to be divergent if it is not convergent. 2.2. Definition [3] Let 𝑓(𝑧, 𝑑) ∈ β„‚[𝑧][[𝑑]]. Define the convergence set of 𝑓 to be Conv( f) = {z ∈ β„‚ : f(z,t) converges in t}. 2.3. Definition [3] A subset 𝐸 ∈ β„‚ is said to be a convergence set if there exists an 𝑓 ∈ β„‚[𝑧][[𝑑]] such that E = Conv(f). Moreover we proved the following theorem, 2.4. Theorem [3] Let 𝑆 = {𝑧1, 𝑧2 , … } be a countable infinite subset of β„‚. Define an F ∈ β„‚[z][[t]] by 𝑭(𝒛, 𝒕) = οΏ½π‘ͺ𝒏[ οΏ½(𝒛 βˆ’ 𝒛𝒋)] 𝒏 𝒋=𝟏 ∞ 𝒏=𝟎 𝒕𝒏, Where 𝐢𝑛 = ( 𝑛 𝛾𝑛� )𝑛, and πœΈπ’ = 𝐦𝐒𝐧 οΏ½ 𝟏 𝟐 𝐦𝐒𝐧 πŸβ‰€π’‹β‰€π’+𝟏 οΏ½π’›π’Š βˆ’ 𝒛𝒋�, 𝟏 𝒏⁄ οΏ½. Then Conv(F) = 𝑆. As a corollary from this theorem we mentioned that the set of rational numbers is a convergence set, we discuss the proof in this paper. 2.5. Corollary The set β„š of rational numbers is a convergence set. Proof. πΉπ‘œπ‘Ÿ 𝑛 ∈ β„•, by Weierstrass theorem see [10] any function on the closed disc {𝑧 ∈ β„‚ ∢ |𝑧| ≀ 𝑛} can be approximated arbitrarily by a polynomial. So for (𝑛𝑛 sin 𝑛!πœ‹π‘§), n πœ–β„•, one can find a polynomial 𝑃𝑛(z) such that |𝑷𝒏(𝒛) βˆ’ 𝒏𝒏 𝐬𝐒𝐧 𝒏!𝝅𝒛)|< 𝟏 𝒏 , for |𝒛| ≀ 𝒏. 170 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 11, No 1, pp 169-172 Let the formal series 𝑭(𝒛, 𝒕) = βˆ‘ 𝑷𝒏(𝒛)π’•π’βˆž 𝒏=𝟎 . Now for every 𝑧 ∈ β„‚ β„š,⁄ the bounded sequence {sin𝑛!πœ‹π‘§)}𝑛=1 ∞ is divergent as 𝑛 extends to infinity. Suppose there exist a sub-sequence of �𝑛𝑗�𝑗=1 ∞ such that limπ‘—β†’βˆž sin 𝑛!πœ‹π‘§ = πœ†, where πœ† is a complex number. Using the previous inequality we get |𝑷𝒏𝒋(𝒛) βˆ’ 𝒏𝒋 𝒏𝒋 𝐬𝐒𝐧 𝒏𝒋!𝝅𝒛)|< 𝟏 𝒏𝒋 , for j ≀ |𝒛|, Which gives that |𝑷𝒏𝒋(𝒛)| β‰₯ |𝒏𝒋 𝒏𝒋 𝐬𝐒𝐧 𝒏𝒋!𝝅𝒛| - 𝟏 𝒏𝒋 , βˆ€ j β‰₯ |𝒛|. |𝑷𝒏𝒋(𝒛) β‰₯ 𝟏 𝟐 |𝝀|𝒏𝒋 𝒏𝒋, For 𝑛𝑗 large enough, 𝐹(𝑧, 𝑑) is divergent βˆ€π‘› ∈ β„•. On the other hand, for x∈ β„š let x = 𝑙 π‘š where 𝑙,π‘š ∈ β„€ and the gcd(𝑙,π‘š) = 1. Then for 𝑛 > π‘š |𝑷𝒏(𝒙)| ≀ | 𝒏𝒏 𝐬𝐒𝐧 𝒏!𝝅𝒙) + 𝟏 𝒏 | = 𝟎 + 𝟏 𝒏 . So 𝐹(π‘₯, 𝑑) is convergence. The Corollary indicates that the whole rational numbers β„š is a convergence set. Now Since the unit interval [0, 1] is compact and simply connected set it's a convergence set. In [3] we proved that a finite intersection of convergence sets is a convergence set, so the intersection of the β„š and the unit interval is a convergence set. 2.6. Example The set of rational number in the unit interval, K = [0, 1] \ β„š, is a convergence set. 3. Conclusion In this paper we find that β„š is a convergence set, which implies that β„š is a quasi-simply-connected set, looking for new sets which is convergence and have other formal power series properties. 171 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2015) Volume 11, No 1, pp 169-172 References [1] S.S. Abhyankar, T.T. Moh, ''A reduction theorem for divergent power series'', J. Reine Angew. Math., 241(1970), pp 27-33. [2] B.L. Fridman, D. 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