Closed-Form Evaluation of Arctanh Power Sums via Infinite Products
Abstract
We establish closed-form expressions for the infinite series sum from n=2 to infinity of arctanh(n^-k) for all integers k >= 2 by connecting these sums to infinite product formulas involving the gamma function. Our approach uses logarithmic manipulations, the Fubini-Tonelli theorem, and Frullani's integral theorem. As applications, we derive a structural identity relating the Riemann zeta function zeta(k) to these sums, establish a new series representation for the Euler-Mascheroni constant gamma, and show that this representation admits an exponentially convergent reformulation via zeta values. We further prove that h(k) = sum from n=2 to infinity of arctanh(n^-k) is strictly decreasing and strictly convex in k, and we establish explicit two-sided bounds and asymptotic expansions. The decimal expansions of the closed-form values and several auxiliary sequences arising from these identities are cataloged in the OEIS.
Keywords
Cite
@article{arxiv.2602.06244,
title = {Closed-Form Evaluation of Arctanh Power Sums via Infinite Products},
author = {Ryan Goulden},
journal= {arXiv preprint arXiv:2602.06244},
year = {2026}
}
Comments
22 pages, v2 removed ambiguity from theorem 1 proof, v3 Revised and expanded manuscript adding a discussion on the arithmetic properties of h(k) for natural k, v4 fixed incorrect OEIS reference for h(2) as well as a serious error in the Cantrell product formula