My problem: Prove that the series $\sum\limits_{n=0}^\infty e^{n(\sin(nx)-2)}$ converges for all $x\in\mathbb{R}$ to a continuous function.
By the root test it converges, but as far as the continuous function part I'm not sure. I mean wouldn't e to any power be continuous no matter the term? Sorry, I'm not the best person around for analysis. But I'm working on it!
The easiest way is to use the Weirestrass M-test. The $n$-th term is uniformly bounded by $e^{-n}$, which is summable. Thus, the sum converges uniformly.
Each term of the sum is continuous, so you have a uniform limit of continuous functions, and therefore the limit function is continuous.