somebody know how to calcualte the following series
I think that is semi convergent
The sum does not converge at all...
Since this has a relatively mild divergence one would hope that the Abel sum of this would exist, and some numerical experiments suggest that it does have an Abel sum for at least some values of a.
Wolfram Alpha fails to find an Abel summation or a Lindelöf sum, but that does not mean that they don't exist, since your question appears to have been written in Mathematica syntax have you tried implementing your preferred summation method in that?
one time a read a book of divergent series of Mr Hardy
and a found a sum
it is possible found it using mclauring Euler i try to extend the sum for other values.
bur wolfram alpha only give the partial sum
sorry for my english lenguage.