Lehmer's Interesting Series

Abstract

The series Sk(z) = ∑ ∞ m=1 [ C2m m ]−1 mk zm is evaluated in a nonrecursive and closed process. It can be analytically continued beyond its domain of convergence |z| < 4 when k = 0, 1, 2, . . . . From this we provide a firm basis for Lehmer’s observation that π emerges from the limiting behavior of Sk(2) as k →∞. 
DOI: 10.4169/amer.math.monthly.120.02.116

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