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!