Prove that the series $\sum_{n=1}^\infty \frac{3n^2+x^4\cos(nx)}{n^4+x^2}$ converges.

Question

Prove that the series $\sum_{n=1}^\infty \frac{3n^2 + x^4\cos(nx)}{n^4+x^2}$ converges to a continuous function $f :\mathbb R \rightarrow \mathbb R$

My attempt: Let $f_n(x)=\frac{3n^2+x^4\cos(nx)}{n^4+x^2}$

$|f_n(x)|\leq\frac{3n^2+x^4}{n^4}$. Let $a_n=\frac{3n^2+x^4}{n^4}$ and $b_n=\frac{1}{n^2}$.

$\lim_{n\rightarrow \infty} \frac{a_n}{b_n}=3$. By comparison test both the series converge or diverge together.

By $p$ test $ b_n=\frac{1}{n^2}$ is convergent. This implies $a_n$ is convergent.

By WM test $\sum f_n(x)$ converges uniformly. We know that it converges to a continuous function.

Answer 1

I think you should localize to an arbitrary fixed closed interval $[a,b]$, and then apply Weierstrass M-Test, the upper bound is then $\dfrac{3n^{2}+b^{4}}{n^{4}}$, there should not be involved with any $x$.

So it converges uniformly on $[a,b]$ and hence the series is continuous on $[a,b]$, and then one can extend $[a,b]$ to include every point to be a continuity point.