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Lehmer’s Interesting Series

Freeman J. Dyson, Norman E. Frankel and M. Lawrence Glasser
The American Mathematical Monthly
Vol. 120, No. 2 (February 2013), pp. 116-130
DOI: 10.4169/amer.math.monthly.120.02.116
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Page Count: 15
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Lehmer’s Interesting Series
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Abstract The series \documentclass{article} \pagestyle{empty}\begin{document} $S_k(z) = \sum_{m=1}^{\infty} \left[C^{2m}_{m}\right]^{-1} m^k z^m$ \end{document} is evaluated in a nonrecursive and closed process. It can be analytically continued beyond its domain of convergence \documentclass{article} \pagestyle{empty}\begin{document} $|z|<4$ \end{document} 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 → ∞.

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