Efficient approximations of finite and infinite real alternating \(p\)-series

Volume 11, Issue 11, pp 1250--1261 http://dx.doi.org/10.22436/jnsa.011.11.05 Publication Date: September 03, 2018       Article History

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


MSC


References