# Efficient approximations of finite and infinite real alternating $p$-series

Volume 11, Issue 11, pp 1250--1261
Publication Date: September 03, 2018 Submission Date: May 02, 2018 Revision Date: July 30, 2018 Accteptance Date: August 03, 2018
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### Authors

Vito Lampret - University of Ljubljana, Slovenia, EU.

### Abstract

For $n\in\mathbb{N}$ and $p\in\mathbb{R}$ the $n$th partial sum of the alternating $p$-series, known also as alternating generalized harmonic number of order $p$, $H^*(n,p):=\sum_{i=1}^n(-1)^{i+1}\frac{1}{i^p}$ is given in the form $H^*(n,p)=S_q(k,n,p)+r^*_q(k,n,p),$ where $k,q\in\mathbb{N}$ with $k<\lfloor n/2\rfloor$ are parameters, controlling the magnitude of the error term $r^*_q(k,n,p)$. The function $S_q(k,n,p)$ consists of $2(k+1)+q$ simple summands and $r^*_q(k,n,p)$ is estimated for $q>-p+1$, as \begin{equation*} \big|r^*_q(k,n,p)\big| < \frac{|p|(|p|+1)\cdots(|p|+q-1)\pi^{p+1}}{3(p+q-1)(2k\pi)^{p+q-1}} . \end{equation*} Additionally, for $p\in\mathbb{R}^+$ and $k,q\in\mathbb{N}$, we have \begin{equation*} \left|r_q^*(k,\infty,p)\right| \le\frac{p(p+1)\cdots(p+q-2)\pi^{p+1}}{3(2k\pi)^{p+q-1}}. \end{equation*}

### Keywords

• Alternating
• alternating generalized harmonic number
• approximation
• estimate
• alternating $p$-series

•  11Y60
•  11Y99
•  33F05
•  33E20
•  40A25
•  41A60
•  65B10
•  65B15

### References

• [1] E. K. Abalo, K. Y. Abalo, Convergence of p-series revisited with applications, Int. J. Math. Math. Sci., 2006 (2006), 8 pages.

• [2] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: with formulas, graphs, and mathematical tables, Dover Publications, New York (1972)

• [3] E. Chlebus, An approximate formula for a partial sum of the divergent p-series, Appl. Math. Lett., 22 (2009), 732–737.

• [4] J. Choi, H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Modelling, 54 (2011), 2220–2234.

• [5] D. Cvijović, The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers, Appl. Math. Comput., 215 (2010), 4040–4043.

• [6] G. Dattoli, H. M. Srivastava , A note on harmonic numbers, umbral calculus and generating functions, Appl. Math. Lett., 21 (2008), 686–693.

• [7] Y. Hansheng, B. Lu, Another proof for the p-series test, College Math. J., 36 (2005), 235–237.

• [8] T. Kim, Y.-H. Kim, D.-H. Lee, D.-W. Park, Y. S. Ro, On the alternating sums of powers of consecutive integers, Proc. Jangjeon Math. Soc., 8 (2005), 175–178.

• [9] V. Lampret, Accurate double inequalities for generalized harmonic numbers, Appl. Math. Comput., 265 (2015), 557–567.

• [10] V. Lampret , Approximating real Pochhammer products: A comparison with powers, Cent. Eur. J. Math., 7 (2009), 493– 505.

• [11] V. Lampret , Asymptotic inequalities for alternating harmonics, Bull. Math. Sci., 2017 (2017), 8 pages.

• [12] V. Lampret, Even from Gregory-Leibniz series $\pi$ could be computed: an example of how convergence of series can be accelerated, Lect. Mat., 27 (2006), 21–25.

• [13] V. Lampret, Wallis’ sequence estimated accurately using an alternating series, J. Number Theory, 172 (2017), 256–269.

• [14] D. A. MacDonald, A note on the summation of slowly convergent alternating series, BIT, 36 (1996), 766–774.

• [15] R. Meštrović, Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun, Int. J. Number Theory, 8 (2012), 1081–1085.

• [16] T. M. Rassias, H. M. Srivastava, Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers , Appl. Math. Comput., 131 (2002), 593–605.

• [17] A. Sîntămărian, Sharp estimates regarding the remainder of the alternating harmonic series, Math. Inequal. Appl., 18 (2015), 347–352.

• [18] A. Sofo, New families of alternating harmonic number sums, Tbilisi Math. J., 8 (2015), 195–209.

• [19] A. Sofo, Polylogarithmic connections with Euler sums , Sarajevo J. Math., 12 (2016), 17–32.

• [20] A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory, 154 (2015), 144–159.

• [21] A. Sofo, H. M. Srivastava, A family of shifted harmonic sums, Ramanujan J., 37 (2015), 89–108.

• [22] Z.-W. Sun , Arithmetic theory of harmonic numbers , Proc. Amer. Math. Soc., 140 (2012), 415–428.

• [23] L. Tóth, J. Bukor, On the alternating series $1 - \frac{1}{2} + \frac{1}{ 3} - \frac{1}{ 4} + ...$, J. Math. Anal. Appl., 282 (2003), 21–25.

• [24] Wolfram, Mathematica, Version 7.0, Wolfram Research, Inc., (1988–2009.),

• [25] T.-C. Wu, S.-T. Tu, H. M. Srivastava , Some combinatorial series identities associated with the digamma function and harmonic numbers, Appl. Math. Lett., 13 (2000), 101–106.

• [26] D.-Y. Zheng , Further summation formulae related to generalized harmonic numbers, J. Math. Anal. Appl., 335 (2007), 692–706.