# Uniform Convergence Infinite Series

I have been given the following series: $$\sum_{n=1}^\infty e^{-n^2x^2}$$ Let now $$a>0$$. Argue that the series converge uniformly on the interval $$[a,\infty)=\{x\in\mathbb{R}:x\geq a\}$$. To do this i have been using Weierstrass' M-test. First i have said that as the exponential function grows faster than the power function the following should be true: $$\frac{1}{e^{n^2x^2}}\leq \frac{1}{n^2}$$. We know that $$\sum_{n=1}^\infty \frac{1}{n^2}$$ is convergent which makes it a convergent majorant series. Pr. Weierstrass' M-test the series $$\sum_{n=1}^\infty e^{-n^2x^2}$$ is then uniformly convergent. Is this approach okay?

You have to prove that your upper bound is valid. Use the fact that $$e^{x} \geq x$$ for $$x >0$$ (which follows from the Taylor expansion). Now $$e^{n^{2}x^{2}} \geq e^{n^{2}a^{2}} \geq n^{2}a^{2}$$ so $$\frac 1 {e^{n^{2}x^{2}}} \leq \frac 1 {n^{2}a^{2}}$$, now compare with the series $$\sum \frac 1 {n^{2}a^{2}}$$ which is convergent.