Uniform convergence of $\sum\limits_{n=1}^{\infty}\frac{n^{2}x^{2}}{e^{n^{2}|x|}}$

Question

I want to study the uniform convergence of $F(x):=\sum\limits_{n=1}^{\infty}\frac{n^{2}x^{2}}{e^{n^{2}|x|}}$ on $S=\mathbb{R}$.

For the root test, it's clear that $F(x)$ exists for all $x\in \mathbb{R}$. But I don't see if the convergence is or not uniform.

Answer 1

$e^x>x^2$ for $x\in R_+$ and $n^2|x|\in R_+$, therefore we have that: $$e^{n^2|x|}>n^4x^2$$ $$\frac{n^2x^2}{e^{n^2|x|}}<\frac{n^2x^2}{n^4x^2}=\frac1{n^2}$$ Since $\sum \frac1{n^2}$ converges, from Weierstrass's test for uniform convergence it follows that your series converges uniformly.