Interchanging integration and summation vs. convergence
I'm trying to calculate the series (all numerical data is calculated in Mathematica)
I proceeded as follows. First the partial fraction decomposition of the summand,
then introducing the variable x:
This is then recognized as the integral
This is still numerically equal to the first expression. Now, I want to interchange the summation and integration and sum the geometric series,
but this integral is numerically equal to .
Why aren't I allowed to sum the geometric series?