오늘의 계산 60 - An Euler Sum

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

 

Proposition. The following holds:[1]

$$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}\frac{H_n^2}{n^2}=\frac{17{\pi }^4}{360}.\end{array}$$

 

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 $\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{H_n}{n}\right)}^2$ - 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.