As part of preparing for an exam, I would like to confirm with you my solution to the exercise below (if you can prove it in a different way, I would like to know).
Thank you!
Question
Is $\sum_{n=1}^{\infty}n2^{-\frac{n^2+x^2}{n} }\sin{(nx)}$ uniformly convergent for $x\in \mathbb{R}$?
My attempt
Let $x\in \mathbb{R}$. Define $u_n(x)=n2^{-\frac{n^2+x^2}{n} }\sin{(nx)}$. Notice that: $$|u_n(x)|= | n2^{-\frac{n^2+x^2}{n} }\sin{(nx)}| \le \left| n \frac{1}{2^{n+\frac{x^2}{n}}}\right| \le n \frac{1}{2^{n}} $$
From WeierstrassM-Test we finish the proof.
Is there a good way to guess if it's convergent or not in advance?