<s>[INST] Prove that $ \sum_{n=1}^{\infty} f_n(x)= \sum_{n=1}^{\infty} \frac {1} {n^2 x^2 +1} $ is convergent How do I prove that $ \sum_{n=1}^{\infty} f_n(x)= \sum_{n=1}^{\infty} \frac {1} {n^2 x^2 +1} $ is convergent for every x in real numbers except for $x=0$? 
I tried using the ratio test, but it doesn't seem to be conclusive. 
 [/INST] You could use the limit comparison test.
Use $\displaystyle b_n = \frac{1}{n^2}$
Let $\displaystyle a_n = \frac{1}{n^2x^2 + 1}$
$\displaystyle \frac{a_n}{b_n} = \frac{n^2}{n^2x^2 + 1}$
$x$ is a constant $x \ne 0$
$\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = \frac{1}{x^2}$
Provided $x \ne 0, x^2 > 0, \frac{1}{x^2} > 0$ we find that since $\displaystyle \zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2}$ converges to $\displaystyle \frac{\pi^2}{6}$ we find that $\sum a_n$ converges as well. (By Limit Comparison Test)
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