How to prove the identity $\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$?

Asked Feb 14 '16 at 22:17

Active Feb 14 '16 at 22:17

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Prove That

$$\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$$

I encountered this identity while reading the article about Harmonic Number on Wikipedia. I thought of using the integral representation of Harmonic Number, but the square is creating trouble.

Any help will be appreciated.
Thanks.

edited Jun 12 '20 at 10:38

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asked Feb 14 '16 at 22:17

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Have you checked the provided reference (from the linked Wolfram article): On an Intriguing Integral and Some Series Related to $\zeta(4)$, David Borwein and Jonathan M. Borwein. Proceedings of the American Mathematical Society, Vol. 123, No. 4 (Apr., 1995), pp. 1191-1198. – Clement C. Feb 14 '16 at 22:27

How to prove the identity $\sum_{n=1}^{\infty} \dfrac{{H_{n}}^2}{n^2} = \dfrac{17}{360} {\pi}^4$?

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