Closed-Form Evaluation of Arctanh Power Sums via Infinite Products
Ryan Goulden
TL;DR
This work derives closed-form evaluations for h(k)=∑_{n=2}^∞ arctanh(n^{−k}) (k≥2) by linking the sums to infinite products f(k) and g(k) that admit gamma- or trig-based closed forms. A key result is h(k)=1/2 ln(f(k)/(2 g(k))) and its equivalent zeta-series h(k)=∑_{m≥0}(ζ((2m+1)k)−1)/(2m+1), together with a structural identity ζ(k)=1+h(k)−C(k) where C(k) decays rapidly. The paper also derives a new, exponentially convergent representation for the Euler–Mascheroni constant γ via Frullani-type integrals and the correction function C(k), and analyzes the analytic properties of h(k) including monotonicity, convexity, and asymptotics. Accelerated computation and precise tail bounds enable efficient evaluation and accurate γ and ζ estimates, with extensive numerical examples and OEIS entries. Overall, the results provide a powerful bridge between arctanh power sums, gamma- and zeta-analytic structures, and fast-converging constants representations with practical computational benefits.
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.