Ratio Mathematica Vol. 34, 2018, pp. 5–13 ISSN: 1592-7415 eISSN: 2282-8214 Sums of Generalized Harmonic Series with Periodically Repeated Numerators (a, b) and (a, a, b, b) Radovan Potůček∗ Received: 01-15-2018. Accepted: 05-04-2018. Published: 30-06-2018. doi:10.23755/rm.v34i0.405 c©Radovan Potůček Abstract This paper deals with certain generalization of the alternating harmonic se- ries – the generalized convergent harmonic series with periodically repeated numerators (a, b) and (a, a, b, b). Firstly, we find out the value of the nu- merators b of the first series, for which the series converges, and determine the formula for the sum s(a) of this series. Then we determine the value of the numerators b of the second series, for which this series converges, and derive the formula for the sum s(a, a) of this second series. Finally, we ver- ify these analytically obtained results and compute the sums of these series by using the computer algebra system Maple 16 and its basic programming language. Keywords: harmonic series, alternating harmonic series, sequence of partial sums, computer algebra system Maple. 2010 AMS subject classifications: 40A05, 65B10. ∗Department of Mathematics and Physics, Faculty of Military Technology, University of De- fence in Brno, Brno, Czech Republic; Radovan.Potucek@unob.cz 5 Radovan Potůček 1 Introduction Let us recall the basic terms, concepts and notions. For any sequence {ak} of numbers the associated series is defined as the sum ∞∑ k=1 ak = a1 + a2 + a3 + · · · . The sequence of partial sums {sn} associated to a series ∞∑ k=1 ak is defined for each n as the sum of the sequence {ak} from a1 to an, i.e. sn = a1 + a2 + · · · + an . The series ∞∑ k=1 ak converges to a limit s if and only if the sequence of partial sums {sn} converges to s, i.e. lim n→∞ sn = s. We say that the series ∞∑ k=1 ak has a sum s and write ∞∑ k=1 ak = s. The harmonic series is the sum of reciprocals of all natural numbers (except zero), so this is the series ∞∑ k=1 1 k = 1 + 1 2 + 1 3 + · · ·+ 1 k + · · · . The divergence of this series can be easily proved e.g. by using the integral test or the comparison test of convergence. In this paper we will deal with the series of the form ∞∑ k=1 ( a 2k −1 + b 2k ) and ∞∑ k=1 ( a 4k −3 + a 4k −2 + b 4k −1 + b 4k ) , where a,b are such numbers that these series converge. If we know, these two types of infinite series has not yet been studied in the literature. The author has previously in the papers [1], [2], and [3] dealt with the series of the form ∞∑ k=1 ( 1 2k −1 + a 2k ) , ∞∑ k=1 ( 1 3k −2 + 1 3k −1 + a 3k ) , ∞∑ k=1 ( 1 3k −2 + a 3k −1 + b 3k ) , and ∞∑ k=1 ( 1 4k−3 + a 4k−2 + b 4k−1 + c 4k ) , so that this contribution is a free follow-up to these three papers. Let us note that in the previous issues of Ratio Mathematica, for example, papers [4] and [5] also deal with the topic infinite series and their convergence. 6 Sums of Generalized Harmonic Series with Periodically Repeated Numerators 2 The sum of generalized harmonic series with pe- riodically repeated numerators (a, b) We deal with the numerical series of the form ∞∑ k=1 ( a 2k −1 + b 2k ) = a 1 + b 2 + a 3 + b 4 + a 5 + b 6 + · · · , (1) where a,b ∈ R are appropriate constants for which the series (1) converges. This series we shall call generalized harmonic series with periodically repeated nu- merators (a,b). We determine the value of the numerators b, for which the series (1) converges, and the sum s(a) of this series. The power series corresponding to the series (1) has evidently the form ∞∑ k=1 ( ax2k−1 2k −1 + bx2k 2k ) = ax 1 + bx2 2 + ax3 3 + bx4 4 + ax5 5 + bx6 6 + · · · . (2) We denote its sum by s(x). The series (2) is for x ∈ (−1,1) absolutely convergent, so we can rearrange it and rewrite it in the form s(x) = a ∞∑ k=1 x2k−1 2k −1 + b ∞∑ k=1 x2k 2k . (3) If we differentiate the series (3) term-by-term, where x ∈ (−1,1), we get s′(x) = a ∞∑ k=1 x2k−2 + b ∞∑ k=1 x2k−1. (4) After reindexing and fine arrangement the series (4) for x ∈ (−1,1) we obtain s′(x) = a ∞∑ k=0 x2k + bx ∞∑ k=0 x2k, that is s′(x) = (a + bx) ∞∑ k=0 ( x2 )k . (5) When we summate the convergent geometric series on the right-hand side of (5) with the first term 1 and the ratio x2, where ∣∣x2∣∣ < 1, i.e. for x ∈ (−1,1), we get s′(x) = a + bx 1−x2 . 7 Radovan Potůček We convert this fraction using the CAS Maple 16 to partial fractions and get s′(x) = a− b 2(x + 1) − a + b 2(x−1) = a− b 2(1 + x) + a + b 2(1−x) , where x ∈ (−1,1). The sum s(x) of the series (2) we obtain by integration in the form s(x) = ∫ ( a− b 2(1 + x) + a + b 2(1−x) ) dx = a− b 2 ln(1 + x)− a + b 2 ln(1−x) + C. From the condition s(0) = 0 we obtain C = 0, hence s(x) = a− b 2 ln(1 + x)− a + b 2 ln(1−x) . (6) Now, we will deal with the convergence of the series (2) in the right point x = 1. After substitution x = 1 to the power series (2) – it can be done by the extended version of Abel’s theorem (see [6], p. 23) – we get the numerical series (1). By the integral test we can prove that the series (1) converges if and only if b + a = 0. After simplification the equation (6), where b = −a, we have s(x) = a + a 2 ln(1 + x) = a ln(1 + x) . For x = 1 and after re-mark s(1) as s(a), we obtain a very simple formula s(a) = a ln 2 , (7) which is consistent with the well-known fact that the sum of the alternating har- monic series ∞∑ k=1 (−1)k+1 k = 1− 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + · · · is equal to ln 2. 3 The sum of generalized harmonic series with pe- riodically repeated numerators (a, a, b, b) Now, we deal with the numerical series of the form ∞∑ k=1 ( a 4k −3 + a 4k −2 + b 4k −1 + b 4k ) = a 1 + a 2 + b 3 + b 4 + · · · , (8) where a,b ∈ R are appropriate constants for which the series (8) converges. This series we shall call generalized harmonic series with periodically repeated nu- merators (a,a,b,b). We determine the value of the numerators b, for which the series (8) converges, and the sum s(a,a) of this series. 8 Sums of Generalized Harmonic Series with Periodically Repeated Numerators The power series corresponding to the series (8) has evidently the form ∞∑ k=1 ( ax4k−3 4k −3 + ax4k−2 4k −2 + bx4k−1 4k −1 + bx4k 4k ) = ax 1 + ax2 2 + bx3 3 + bx4 4 + · · · . (9) We denote its sum by s(x). The series (9) is for x ∈ (−1,1) absolutely convergent, so we can rearrange it and rewrite it in the form s(x) = a ∞∑ k=1 x4k−3 4k −3 + a ∞∑ k=1 x4k−2 4k −2 + b ∞∑ k=1 x4k−1 4k −1 + b ∞∑ k=1 x4k 4k . (10) If we differentiate the series (10) term-by-term, where x ∈ (−1,1), we get s′(x) = a ∞∑ k=1 x4k−4 + a ∞∑ k=1 x4k−3 + b ∞∑ k=1 x4k−2 + b ∞∑ k=1 x4k−1. (11) After reindexing and fine arrangement the series (11) for x ∈ (−1,1) we obtain s′(x) = a ∞∑ k=0 x4k + ax ∞∑ k=0 x4k + bx2 ∞∑ k=0 x4k + bx3 ∞∑ k=0 x4k, that is s′(x) = (a + ax + bx2 + bx3) ∞∑ k=0 ( x4 )k . (12) After summation the convergent geometric series on the right-hand side of (12) with the first term 1 and the ratio x4, where ∣∣x4∣∣ < 1, i.e. for x ∈ (−1,1), we get s′(x) = a + ax + bx2 + bx3 1−x4 = (a + bx2)(1 + x) (1 + x2)(1−x)(1 + x) = a + bx2 (1 + x2)(1−x) . We convert this fraction using the CAS Maple 16 to partial fractions and get s′(x) = a + b 2(1−x) + a− b + ax− bx 2(1 + x2) , where x ∈ (−1,1). The sum s(x) of the series (9) we obtain by integration: s(x) = ∫ ( a + b 2(1−x) + a− b 2(1 + x2) + (a− b)x 2(1 + x2) ) dx = = − a + b 2 ln(1−x) + a− b 2 arctanx + a− b 4 ln(1 + x2) + C. 9 Radovan Potůček From the condition s(0) = 0 we obtain C = 0, hence s(x) = − a + b 2 ln(1−x) + a− b 2 arctanx + a− b 4 ln(1 + x2). (13) Now, we will deal with the convergence of the series (9) in the right point x = 1. After substitution x = 1 to the power series (9) we get the numerical series (8). By the integral test we can prove that the series (8) converges if and only if a + b = 0. After simplification the equation (13), where b = −a, we have s(x) = aarctanx + a 2 ln(1 + x2) = a 2 [ 2 arctanx + ln(1 + x2) ] . For x = 1 and after re-mark s(1) as s(a,a), we obtain a simple formula s(a,a) = a 4 (π + 2 ln 2) . (14) 4 Numerical verification We have solved the problem to determine the sums s(a) and (a,a) above of the convergent numerical series (1) and (8) for several values of a (and for b = −a) by using the basic programming language of the computer algebra system Maple 16. They were used two following very simple procedures sumab and sumaabb: sumab=proc(t,a) local r,k,s; s:=0; r:=0; for k from 1 to t do r:=a*(1/(2*k-1)- 1/(2*k)); s:=s+r; end do; print("s(",a,")=",evalf[9](s), "f=",evalf[9](a*ln(2))); end proc: sumaabb:=proc(t,a) local r,k,s; s:=0; r:=0; for k from 1 to t do r:=a*(1/(4*k-3)+ 1/(4*k-2)- 1/(4*k-1)- 1/(4*k)); s:=s+r; end do; print("s(",a,a,")=",evalf[9](s), "f=",evalf[9](a*(Pi+2*ln(2))/4); end proc: 10 Sums of Generalized Harmonic Series with Periodically Repeated Numerators For evaluation the sums s(106,a) and s(106,a,a) and the corresponding val- ues s(a) and s(a,a) defined by the formulas (7) and (14) it was used this for-loop statement: for a from 1 to 10 do sumab(1000000,a); sumaabb(1000000,a); end do; The approximative values of the sums s(106,a), s(a), s(106,a,a), and s(a,a) rounded to seven decimals obtained by these procedures are written into the fol- lowing Table 1. Let us note that the computation of 20 values s(a) and s(a,a) took almost 51 hours 26 minutes. The relative quantification accuracies r(a) = |s(106,a)−s(a)| s(106,a) of the sum s(a,106) and r(a,a) = |s(106,a,a)−s(a,a)| s(106,a,a) of the sum s(a,a,106) are stated in the fourth and eighth columns of Table 1. These relative quantification accuracies are approximately between 4 ·10−7 and 2 ·10−7. a s(106,a) s(a) r(a) a s(106,a,a) s(a,a) r(a,a) 1 0.6931469 0.6931472 4·10−7 1 1.1319715 1.1319718 3·10−7 2 1.3862939 1.3862944 4·10−7 2 2.2639430 2.2639435 2·10−7 3 2.0794408 2.0794415 3·10−7 3 3.3959145 3.3959153 2·10−7 4 2.7725877 2.7725887 4·10−7 4 4.5278860 4.5278870 2·10−7 5 3.4657347 3.4657359 3·10−7 5 5.6598575 5.6598588 2·10−7 6 4.1588816 4.1588831 4·10−7 6 6.7918290 6.7918305 2·10−7 7 4.8520285 4.8520303 4·10−7 7 7.9238005 7.9268023 2·10−7 8 5.5451754 5.5451774 4·10−7 8 9.0557720 9.0557740 2·10−7 9 6.2383224 6.2383246 4·10−7 9 10.1877435 10.1877458 2·10−7 10 6.9314693 6.9314718 4·10−7 10 11.3197150 11.3197175 2·10−7 Table 1: The approximate values of the sums of the generalized harmonic series with periodically repeating numerators (a,−a) and (a, a,−a,−a) for a = 1, 2, . . . , 10 5 Conclusions In this paper we dealt with the generalized harmonic series with periodically repeated numerators (a,b) and (a,a,b,b), i.e. with the series ∞∑ k=1 ( a 2k −1 + b 2k ) = a 1 + b 2 + a 3 + b 4 + a 5 + b 6 + · · · 11 Radovan Potůček with the sum s(a) and with series ∞∑ k=1 ( a 4k −3 + a 4k −2 + b 4k −1 + b 4k ) = a 1 + a 2 + b 3 + b 4 + · · · with the sum s(a,a), where a,b ∈ R. We derived that the only value of the numerators b ∈ R, for which these series converge, are b = −a, and we also derived that the sums of these series are determined by the formulas s(a) = a ln 2 and s(a,a) = a 4 (π + 2 ln 2) . So, for example, the series ∞∑ k=1 ( 5 2k −1 − 5 2k ) = 5 2 ∞∑ k=1 1 (2k −1)k has the sum s(5) . = 3.4657 and the series ∞∑ k=1 ( 5 4k −3 + 5 4k −2 − 5 4k −1 − 5 4k ) = 5 4 ∞∑ k=1 32k2 −24k + 3 (4k −3)(2k −1)(4k −1)k has the sum s(5,5) . = 5.6599. Finally, we verified these two main results by computing some sums by us- ing the CAS Maple 16 and its basic programming language. These generalized harmonic series so belong to special types of convergent infinite series, such as geometric and telescoping series, which sum can be found analytically and also presented by means of a simple numerical expression. From the derived formulas for s(a) and s(a,a) above it follows that a = s(a) ln 2 and a = 4s(a,a) π + 2 ln 2 . These relations allow calculate the value of the numerators a for a given sum s(a) or s(a,a), as illustrates the following Table 2: s(a) a s(a,a) a 1 1.4427 1 0.8834 ln 0.5 . = −0.6391 −1 (−π −2 ln 2)/4 .= −1.1320 −1 ln 2 . = 0.6391 1 (π + 2 ln 2)/4 . = 1.1320 1 Table 2: The approximate values of the numerators a for some sums s(a) and s(a, a) 12 Sums of Generalized Harmonic Series with Periodically Repeated Numerators 6 Acknowledgements The work presented in this paper has been supported by the project ”Rozvoj oblastı́ základnı́ho a aplikovaného výzkumu dlouhodobě rozvı́jených na katedrách teoretického a aplikovaného základu FVT (K215, K217)” VÝZKUMFVT (DZRO K-217). References [1] R. Potůček, Sums of Generalized Alternating Harmonic Series with Period- ically Repeated Numerators (1,a) and (1,1,a). In: Mathematics, Informa- tion Technologies and Applied Sciences 2014 post-conference proceedings of selected papers extended versions. University of Defence, Brno, (2014), 83-88. 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