Problem: Study the possibility of taking the derivative of the following series:
$$\sum_\limits{n=1}^{\infty}\arctan\left(\frac{x}{n^2}\right)\:\:,x\in\mathbb{R}$$
I have studied the following theorem:
Theorem: Suppose that $\sum_\limits{n=k}^{\infty}f_n$ converges uniformly to $F$ on $S=[a,b]$. Assume that $F$ and $f_n\:\:,n\geqslant k$, are integrable on $[a,b]$. Then:
$$\int_\limits{a}^{b}F(x)dx=\sum_\limits{n=k}^{\infty}\int_\limits{a}^{b}f_n(x)dx$$
Following the theorem I would need to check out if the derivative converges uniformly $\sum_\limits{n=1}^{\infty}(\frac{1}{1+\frac{x^2}{n^4}}\frac{2x}{n^4})$ I tried to apply Dirichlet to latter. Once I know that $\sum_\limits{n=1}^{\infty}\frac{2x}{n^4}$ by the integral test converges uniformly. However I was not able to apply it due to the fact that I could not prove $$\sum_\limits{n=1}^{\infty}\left(\frac{1}{1+\frac{x^2}{n^4}}\right)\leqslant M$$
Question:
How should I solve the problem?
How should I prove the series $\sum_\limits{n=1}^{\infty}\arctan\left(\frac{x}{n^2}\right)$ converge?
Thanks in advance!