<s>[INST] Show that the derivative of a series $\sum_{n=1}^\infty \frac{-2x}{(2+x^2)^{n+1}}$ converges uniformly Consider the derivative series
$$
D = \sum_{n=1}^\infty \frac{-2x}{(2+x^2)^{n+1}}
$$
of
$$
S = \sum_{n=1}^\infty \frac{1}{n} \frac{1}{(2+x^2)^n}
$$
I have shown that $S$ converges by the use of Weiterstrass' M-test by saying that
as $x \in \mathbb{R}$ we must have that
$$
\frac{1}{(2+x^2)^n} \leq \frac{1}{n}
$$
for all $n \in \mathbb{N}$. This means that 
$$
\frac{1}{n} \frac{1}{(2+x^2)^n} \leq \frac{1}{n^2}
$$
and from analysis we know that 
$$
\sum_{n=1}^\infty \frac{1}{n^2}
$$
converges. But I am struggling to prove that $D$ converges uniformly. Do I have to use the same approach? I found that for $x \in \mathbb{R}$ and for all $n \in \mathbb{N}$ that 
$$
 \frac{-2x}{(2+x^2)^{n+1}} \leq \frac{1}{2^n}
$$
which is a geometric series that converges. Can I use this? Thanks in advance. 
 [/INST] If $f(x)=\frac{-2x}{(2+x^2)^{n+1}}$, then$$f'(x)=2 \left(x^2+2\right)^{-n-2} \left((2 n+1) x^2-2\right).$$So, the maximum of $|f|$ is attained when $x=\pm\sqrt{\frac2{2n+1}}$ and$$\left|f\left(\pm\sqrt{\frac2{2n+1}}\right)\right|=2 \sqrt{2} \sqrt{\frac{1}{2 n+1}} \left(\frac{2}{2 n+1}+2\right)^{-n-1}.$$Since$$\lim_{n\to\infty}2 \sqrt{2} \sqrt{\frac{1}{2 n+1}} \left(\frac{2}{2 n+1}+2\right)^{-n-1}=0,$$the convergence is uniform by the Weierstrass $M$-test.
</s>