Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$

Question

Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$?

So I started by saying that by the geometric series test where $a=x^2$ and $|r| = |\frac{1}{e^x}| \leq 1$, the series converges pointwise.

But how do I exactly prove that it converges uniformly? I am quite sure it is by weistrass test but I can not find an upper bound to compare it to! Any direction would be appreciated!

Answer 1

Let $$f_n(x)=x^2e^{-nx}$$ then we have $$f'_n(x)=e^{-nx}\left(2x-nx^2\right)=0\iff x=0\ \text{or}\ x=\frac{2}{n}$$ so $$||f_n||_\infty=f_n\left(\frac{2}{n}\right)=\frac{4}{n^2}e^{-2}$$ hence the series $\displaystyle \sum_{n=1}^\infty ||f_n||_\infty$ is convergent and then the series $\displaystyle \sum_{n=1}^\infty f_n$ is uniformly convergent.

Answer 2

You can compute the remainder term explicitly, using the formula for the geometric series: $$ r_N(x) = \sum_{n=N}^\infty x^2 e^{-nx} = x^2e^{-Nx} \sum_{j = 0}^\infty e^{-jx} = \dots $$ Now find the maximum of $r_N$ on $[0, \infty)$. (It's obviously a positive function.)

If this maximum tends to $0$ as $N \to \infty$, then the convergence is uniform. Otherwise it isn't.