show a series is not convergence uniformly

Question

I am trying to prove that the series $$ \sum_{n=1}^\infty f_n(x)=\sum_{n=1}^\infty-x\exp(-n^2 x^2)$$ is not convergence uniformly in $[-1,1]$.

What I usually do when I trying to prove a series is not uniformly convergence is to find a sequence $x_n$ such that $f_n(x_n)$ is greater then some $\epsilon>0$ for all $n$. However, in this case, even I take the maximal point $x_n=1/\sqrt{2}n$, I have $f_n(x_n)=c/n$ for some constant $c$, but it goes to $0$...

what else can I try to prove a series is not uniformly convergence? especially when it is hard to write down the pointwise limit.

Answer 1

Note that

$$\sup_{x}\left|\sum_{k=n+1}^{\infty}xe^{-k^2x^2}\right|\geq\sup_{x}\left|\sum_{k=n+1}^{2n}xe^{-k^2x^2}\right|\geq \sup_{x}|nxe^{-4n^2x^2}|=\frac{e^{-1/2}}{2\sqrt{2}}$$

which does not converge to $0$ as $n \rightarrow \infty$.

So the convergence is not uniform.

Answer 2

Just show that the limit is not continuous. (This will be enough, since the uniform limit of a sequence of continuous functions is continuous.)

You don't need to calculate an explicit formula for the limit-function; just show that it fails to be continuous at one particular point.