I want to compute $$ \sum ^{\infty }_{n=1}\dfrac {\left( 2n+k-2\right) !\zeta \left( 2n\right) \left( -1\right) ^{n-1}}{\left( 2\pi \right) ^{2n}}. $$
This series is surely not convergent for any value of $k$,but moreover it is a divergent series. We shall assign a value to this series by adapted Borel sums. It also grows faster than $n^{2n}$.I need a function depended on "k" This series might be considered from a different angle.I think regularized Borel-sum works ,perhaps you guys give me a little help to simplify this series