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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