# 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
• 739 Views ### 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

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