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New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)

Jonathan Sondow
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Abstract

In math.CA/0211148 we observed that $\ln(4/ \pi)$ is an "alternating" analog of Euler's constant $\gamma$. Here we use the binary expansion of an integer to give a rational series for $\ln(4/ \pi)$ analogous to Vacca's series for $\gamma$. Also, using a known generalization of Vacca's series to other bases, we accelerate Addison's series for $\gamma$. Along the way, we give new proofs of Vacca's and Addison's formulas.

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Jonathan Sondow. 2005-08-01. New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi). https://arxiv.org/abs/math/0508042

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