I'm currently studying sequences and series of functions, and I've begun doing exercises. I've soon found that, while using the definition of uniform convergence is viable for sequences of functions, it is not so much for series of functions. My question is: are there known sufficient conditions for uniform convergence of such series? (If so, I'd be grateful if you could point me to a proof of the tests).
Thank you in advance.
E.G.
$$\sum_{n=0}^{\infty}\frac{n^n}{n!x^n},x\in \mathbb{R^+}$$ I have found that the series converges point-wise for $x>e$ and diverges for $0<x\leq e$. The convergence is obviously absolute but not total, since $\sup_{x\in]e,+\infty[}f_n(x)=\frac{n^n}{n!e^n}$ which diverges using Stirling's formula. I don't know how to proceed with uniform convergence though.