Uniform convergence

Question

I got a task: research $$\sum_{n=1}^\infty~e^{-nx^2}\sin nx$$ for a uniform convergence. I see that $ \sup_{x\in X} |f_n(x)-f(x)|\to 0 $ when $x\ne0$. But what I must do when $x=0$?

Answer 1

If the series were uniformly convergent on $[0,1]$, then we would have $a_n:=\sup_{x\in [0,1]}e^{-nx^2}|\sin(nx)|\to 0$.

But $a_n\geqslant e^{-n\cdot n^{-2}}|\sin(1)|=e^{-1/n}|\sin(1)|\geqslant |\sin(1)|/2$ for $n$ large enough.