I need to clarify why the output gives not an appropriate result ? Actually the code is working without giving error and gives and output. But it stacks with adding exponential function in the sum of series.
I am happy to hear any ways to solve this type of calculation with using exponential integrals in the sum of series form.
Do you have any reason to think there is a closed form for the double sum with ExpIntegralE function?
Most Sum don't have one.
Maybe the best you can do is numerical methods with approximation.