Lehmer's Interesting Series

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 $k=0,1,2,\dot{.}.$. From this we provide a firm basis for Lehmer's observation that $\pi$ emerges from the limiting behavior of $S_k(2)$ as $k\to \infty$.

14 pages

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