I want 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}} $

Question

I want 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 converge for any value of k,but even it is diverge series,we shall assign value for this series by cesaro,abel,borel etc sums but ı could not go any further