I'm currently studying real analysis. We are working with series of various forms, uniform convergence, convergence and divergence for series etc.
I recently met this problem, where one had to evaluate $\int_{0}^{\infty} \frac{x^3}{e^x-1}dx$. In my solution I used the fact that $1/(e^x-1) = \sum_{n=1}^{\infty} e^{-nx}$. Either way, I thought it was a pretty cool problem, since I had to combine different aspects of the course. For instance, I then further down the line had to use Weierstrass M-test to show convergence for the series, in order to interchange the integral with the summation.
My question is then, do you people out there maybe have some good tips on where problems like these can be found. I'd mainly like problems where one needs to show uniform convergence in order to change the order of operations with integrals, sums and limits. I hope you can provide me with some examples / pdfs / links of where I can find such problems. They're really fun :)
Thank you.