Lehmer's Interesting Series

TL;DR

This paper evaluates Lehmer's series in closed form and explores its analytic continuation, providing a foundation for understanding how π arises from the series' limiting behavior as the parameter k approaches infinity.

Contribution

It offers a non-recursive closed-form evaluation and analytic continuation of Lehmer's series, linking it to the emergence of π in the limit as k increases.

Findings

Series evaluated in closed form for all k

Analytic continuation beyond original convergence domain

Connection established between series limit and π

Abstract

The series $$S_k(z)=\sum_{m=1}^{\infty}\frac{m^kz^m}{(\{array}{c} 2m m \{array})}$$ is evaluated in non-recursive closed and analytically continued beyond its domain of convergence $0\le |z|<4$ for $k=0,1,2,\...$. From this we provide a firm basis for Lehmer's observation that $\pi$ emerges from the limiting behavior of $S_k(2)$ as $k\rightarrow\infty$.