uniform convergence of $\sum_{n=1}^{\infty} \frac{n}{e^{nx^2}}$

Question

I have found pointwise convergence in R-{0}.To study uniform convergence I calculate $f'_n(x)$ and I found a max in x=0 so Sup $f_n(x)$= Sup $\frac{n}{e^{nx^2}}=n$ in R-{0} and I can't use Weierstrass theorem. So in which intervals can I study uniform convergence?

Answer 1

For $ x \geq 1$ we have that $e^{nx^2} \geq e^n$ hence $\frac{n}{e^{nx^2}} \leq \frac{n}{e^n}$ and series $\sum_{n=1}^\infty \frac{n}{e^n}$ converges by the ratio test. So the convergence is uniform for $x \geq 1$.