Problem On Infinite Series Yielding A Bunch of Known Constants[Proposed By Cornel Ioan Valean,PDF]

A Cool Infinite Series Yielding A Bunch of Known Constants

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This is another beautiful proposition by Cornel Ioan Valean from Timis, Romania.

The problem was published on August 28, 2017.

Submit your solutions on the comment box below or at $cornel2001\_ro@yahoo.com$.

Infinite Series Yielding A Bunch of Known Constants

Prove that 

$\sum^\infty_{n=1}\frac{(-1)^{n-1}}{n}(1+\frac{1}{2^5}+\cdots+\frac{1}{(2n-1)^5})$

$=\frac{5089}{2048}\zeta(6)-\frac{15}{512}\log(2)\zeta(5)+\frac{15}{8}\zeta(4)G-\frac{9}{1024}\zeta^2(3)-\frac{1}{384}G\psi^{(3)}(\frac{1}{4})$

 

Where $\zeta$ is the Riemann-zeta function, $G$ denotes the Catalan's constant and $\psi^(n)$ represents the Polygamma function.

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