## 오늘의 계산 60 - An Euler Sum

2013. 11. 6. 18:33

Take Home Exam을 풀어야 하는데, 나는 이런 거나 계산하고 있을 뿐이고….

Proposition. The following holds:[1] \begin{align*} \sum_{n=1}^{\infty} \frac{H_{n}^{2}}{n^{2}} = \frac{17\pi^{4}}{360}. \end{align*}

Proof. See the reference [1] below.

Remark. This identity was first conjectured by Enrico Au-Yeung, a student of Jonathan Borwein, using computer search and the PSLQ algorithm, in 1993.[2] It is subsequently solved by the Borweins.[3]

## References

1. Erich, A closed form for the sum $\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$ - Math StackExchange
2. Bailey, David (1997). "New Math Formulas Discovered With Supercomputers". NAS News 2 (24).
3. D. Borwein and J. M. Borwein, `On an intriguing integral and some series related to ζ(4),' Proc. Amer. Math. Soc. 123 (1995), 1191-1198.