I encountered this summation in Statistical Mechanics. $$\sum_{n=1}^{\infty}\frac{1}{n(\sinh(kn))^3}$$ I know this series converges. Can we find its closed form solution? Here k is some positive real number.
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What is the sum over, $k$ or $n$? – Anon123 Mar 20 '18 at 11:51
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2For particular values of $k$, this series has a closed form in terms of elliptic singular values. See Zucker, The Summation of Series of Hyperbolic Functions. See also this related question: https://math.stackexchange.com/questions/304390/summation-of-infinite-series-with-hyperbolic-sine – Jack D'Aurizio Mar 20 '18 at 16:09
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Approximate it. The first term dominates the sum so to first order it's $\frac{1}{\sinh(k)^3}$. This is a perfect approximation for large $k$. For small $k < 1$ it's off by $10%$. – Winther Aug 16 '18 at 13:10