The convergence and partial convergence of alternating series

Abstract:

The alternating series is $\Sigma _{n = 1}^\infty {j_n}f(n) = [j]f$, with f a single-signed monotonic function of the real variable x. The ${j_n}$ are $\pm 1$, their sign fixed by repetition of the ’template’ [j] of finite length 2p. [j] constitutes a difference scheme of ’differential order’ D, which can be determined. The principal theorem is that $[j]f$ is ’partially convergent’ if and only if ${\lim _{x \to \infty }}{f^{(D - 1)}}(x)$ is bounded. A series is partially convergent when the limit as $M \to \infty$ of the sum of 2pM terms exists. For [j] ’pure’, the improved Euler-Maclaurin expansion (IEM) gives the compact representation \begin{equation}\tag {A} S^{(p)} \sim - \frac {\mu (D)}{2p} \sum _{r = 0}^\infty (2p)^{2r} \frac {B_{2r}(1/2)}{(2r!)} f^{(2r+D-1)} (\theta _r),\quad 1 - p \leqslant \theta _r \leqslant p. \end{equation} $S^{(p)}$ is the sum, $\mu (D)$ is the Dth ’template moment’, and the ${B_{2r}}$ are Bernoulli numbers. Efficient means for practical summation of these series follow also from IEM. In illustration, 10 alternating series with D ranging from 1 to 3 are summed using IEM. It is found that the leading term of (A) with $\theta _0 = 1/2$ gives a simple but effective estimate of sums. The paper also gives a comparison with Euler’s transformation in the case $p = 1$ and discusses sums to N terms with $N/2p$ nonintegral and finite but large.