Infinite series identities involving $π$ and $\ln 2$
Aliyah Maxwell-Abrams, Robert Schneider
TL;DR
This paper addresses finding new infinite series identities whose sums are linear combinations of $\pi$ and $\ln 2$ that are not listed in standard references. It builds on prior work connecting $\pi$ to $\ln 2$ through specific series and uses linear combinations and term-wise fraction arithmetic to derive new identities, including six identities and supplementary ones with four- and five-denominator products. The authors illustrate concrete formulas such as $\sum 1/(n(2n-1)(4n-3)) = \pi/3$ and $\pi = 2 + 16 S_{19}$, and discuss convergence properties and computational implications relative to classical series. By expanding the catalog of Euler-type and Ramanujan-style series for $\pi$-related constants, the work invites further discovery guided by historical methods and existing literature.
Abstract
We prove identities for six infinite series whose values involve linear combinations of $π$ and $\operatorname{ln} 2$, that do not appear in standard infinite series references.