Closed-Form Evaluation of Arctanh Power Sums via Infinite Products
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911480878727168 |
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| author | Goulden, Ryan |
| author_facet | Goulden, Ryan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06244 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Closed-Form Evaluation of Arctanh Power Sums via Infinite Products Goulden, Ryan General Mathematics 40A25, 11M06, 33B15 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. |
| title | Closed-Form Evaluation of Arctanh Power Sums via Infinite Products |
| topic | General Mathematics 40A25, 11M06, 33B15 |
| url | https://arxiv.org/abs/2602.06244 |