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

Lampret, Vito ¹

¹ University of Ljubljana, Slovenia, EU

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 11, p. 1250-1261.

Résumé

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*}

Détail

DOI : 10.22436/jnsa.011.11.05

Classification : 11Y60, 11Y99, 33F05, 33E20, 40A25, 41A60, 65B10, 65B15
Keywords: Alternating, alternating generalized harmonic number, approximation, estimate, alternating \(p\)-series

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