We consider the sequence $(f_k)$ defined by $f_k:\mathbb{R}\to\mathbb{R}$, $f_k(x)=\dfrac{x^2}{(1+x^2)^k}$ for each $k\in \mathbb{N}$.
I already proved that $\displaystyle\sum_{k=1}^{\infty}f_k$ converges pointwise in $\mathbb{R}$.
My question is, does $\displaystyle\sum_{k=1}^{\infty}f_k$ converge uniformly?
Thanks.
A full answer to this question (it was already partially answered in the comments):
Define $$ g_n(x)=\sum_{k=1}^{n}f_k(x)\\ =\frac{x^2}{x^2+1} \sum_{k=0}^{n}\frac{1}{(1+x^2)^k}\\ =\frac{x^2}{x^2+1}\frac{1-[1/(1+x^2)]^{n+1}}{1-[1/(1+x^2)]}\\ =\frac{x^2}{x^2+1}\frac{1-[1/(1+x^2)]^{n+1}}{[x^2/(1+x^2)]}\\ =(1-[1/(1+x^2)]^{n+1}) $$
The pointwise limit of this function is $$ g(x)=\begin{cases}1 & x≠0 \\ 0 & x=0 \\ \end{cases} $$ Since this continuous sequence approaches a non-continuous pointwise limit, we may conclude that the sequence does not converge uniformly.
(Thanks for the correction David)