Good day, I have a question to complete which asks me to prove whether the following series is point-wise convergent and uniformly convergent. Obviously if it does not converge point-wise it will not converge uniformly (using the "strength" of uniform convergence).
$$\sum_{n=0}^\infty e^{-xn^2} \text{ on domain } (0,\infty)$$
I have shown that is is point-wise convergent by comparing it to the Geometric series $$\frac{1}{1-e^{-x}}$$
But I am not quite sure how I can go about disproving uniform convergence. I am tempted to use a corollary of $$\ f_n \to f \text{ uniformly } iff\ \forall\epsilon>0 \ \exists N \ \forall n,m \geq N \ \ \Vert f_n-f_m \Vert_{sup}<\epsilon$$ which states that if $$\ \Vert g_n \Vert_{sup} \nrightarrow 0 \ \implies \sum_{n=0}^\infty g_n \text{ does not converge uniformly}$$
But I am not too sure if that is valid.
Best Answer
It's not uniformly Cauchy. For $M<N$ we have $$ \sup_{0<x}\sum_{n=M}^N e^{-n^2x}>\sup_{0<x}e^{-M^2x}=1 $$