Estimating the remainder of an alternating \(p\)-series revisited

Abstract

For the \( n \)th remainder \( R_n(p):=
\sum_{k=n+1}^{\infty}(-1)^{k+1}k^{-p} \) of an alternating
\( p \)-series, several asymptotic estimates are presented. For
example, for any integer \( n \ge 3 \), and \( p \in \mathbb{R}^+ \), we have

\[
R_n(p) = \frac{(-1)^n}{2\left(2\left\lfloor \frac{n+1}{2} \right\rfloor\right)^p} -
\frac{p}{4\left(2\left\lfloor \frac{n+1}{2} \right\rfloor\right)^{p+1}}
+ \varepsilon_n^*(p)
\]

and

\[
\left| \varepsilon_n^*(p) \right| < \frac{p(p+1)}{5\,(n-2)^{p+2}},
\]

where \( \lfloor x \rfloor \) denotes the integer part (the floor) of \( x \).