text: Q:
  
  Convergence of sequence of functions $\sum_{n\ge1 }ne^{-nx^2}$
  
  How to show that the series $$\sum_{n\ge1 }ne^{-nx^2}$$ is not uniformly convergent for $x\in (0,\infty)$ whereas converges uniformly when $x\in (a,\infty),\ a>0$? 
  What I observed is, for $x>a$, $$ ne^{-nx^2}\le ne^{-na^2}. $$  Since $$\sum_{n\ge 1}ne^{-na^2}\le \infty\implies \sum_{n\ge 1}f_n\ \text{converges uniformly when}\ x\in (a,\infty). $$ But, how to do the first case?
  
  A:
  
  Note that$$\sum_{n=1}^\infty ne^{-nx^2}=\sum_{n=1}^\infty n\left(e^{-x^2}\right)^n=\frac{e^{-x^2}}{\left(1-e^{-x^2}\right)^2},$$which is an unbounded function on $(0,+\infty)$. Since each function $\sum_{n=1}^Nne^{-nx^2}$ is bounded there, the convergence cannot be uniform.
  
  