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?