Assuming :

Well the imaginary part of:

if it exists, is zero, since the partial sums of the first series are equal to the conjugates of the partial sums of the second. However since you cannot rearrange conditionally convergent series what I say below about the real part also applies here.

The real part is:

If this is zero (or even converges, since it does not satisfy the conditions of the alternating serier test) I don't know.

(The integral does not converge, but that also means nothing as the conditions for the integral test for convergence are not satisfied by this series)

If I were a betting person I think I would put my money on non-convergence.

(Note is divergent)

It might be interesting to consider other definitions of convergence for this problem. (for some reason I cannot find any references on line for these at present),

but I can givew an example:

For the dubiously convergent series , define the "sum" to be:

RonL