Is the series uniform convergent in $(0,\infty)$?

Question

For $$f(x)=\sum_{n=1}^{\infty}\frac{1}{1+n^2x}$$ And is it bounded in $(0,\infty)$?

Answer 1

Note that

$$\sup_{x}\sum_{k=n+1}^{\infty}\frac{1}{1+k^2x}\geq\sup_{x}\sum_{k=n+1}^{2n}\frac{1}{1+k^2x}\geq \sup_{x}\frac{n}{1+4n^2x}=n$$

which does not converge to $0$. So the convergence is not uniform.

Answer 2

not uniformly convergent, since by picking $x = 1/n^2$ you can get the $n$-th term to be $1/2$ and since $n$ was arbitrary and your series has positive terms this contradicts uniform convergence. it's also not bounded since the terms are decreasing for a fixed $x$. what we've done before shows that you can get a sum bigger than $n/2$ for any $n$.