Prove: $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$ I want to ask you if can it be so simple to prove that $\lim _{x \to \infty}\sum_{1}^{\infty}\frac{x^2}{1+n^2x^2}=\sum_{1}^{\infty}\frac{1}{n^2}$ by divide the numerator and denominator with $x^2$ and that's it? 
If it this simple indeed you can write a comment and I'll delete the question after I'll read it, or perhaps I'm missing something important (and I should involve power series).
Thanks!

In general, interchanging limits and sums or integrals can be tricky.  It's not always true that
$\lim_{x \to \infty} \sum_{n=1}^\infty f_n(x) = \sum_{n=1}^\infty \lim_{x \to \infty} f_n(x)$, even when both sides converge.  It is true for dominated
convergence (if there is some convergent series $\sum_{n=1}^\infty b_n$ with $|f_n(x)| \le b_n$ for all $n$ and $x$) and monotone convergence (if $f_n(x)$ is positive, and increasing as a function of $x$).  This example satisfies both conditions.
