Lehmer's Interesting Series

Details Lehmer's Interesting Series Lehmer's Interesting Series

2010-09-22

arxiv.org/abs/1009.4274v4

Physics

Mathematical Physics

14 pages

Scientific paper

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 $k\rightarrow\infty$.

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