How to prove series of functions converges to a continuous function? The question is to prove that  the series
$$
\sum_{k=1}^\infty\frac{k^2+x^4}{k^4+x^2}
$$
converges to a continuous function $f:\Bbb R\to\Bbb R$. I have tried to prove uniform convergence. Is the series uniformly converges?

Hint: Let $a>0.$ On the interval $[-a,a],$ we have
$$\frac{k^2+x^4}{k^4+x^2} \le \frac{k^2+a^4}{k^4} = \frac{1}{k^2}+ a^4\frac{1}{k^4}.$$
This shows the series converges uniformly on $[-a,a]$ (why?). Since this is true for any $a>0,$ ...
