Find where this series uniformly converges

Question

Given the following series:

$$f_n(x) = \frac{x^2}{(1+x^2)^n}$$

for $x\in\mathbb{R}$, and let $s_k = \sum_{n=0}^kf_n(x)$.

Find values $a < b$ where the series uniformly converges on $[a, b]$.

So far, I found that $s_k(x) = 1+x^2$ as $k\to\infty$.

Answer 1

Let $s(x) = 1 + x^2$, the point-wise limit as $k \to \infty$. We want to find the intervals such that the function $$ |s_k - s| = \left|\sum_{n=k+1}^\infty f_n(x) \right| $$ approaches $0$ uniformly. Note in particular that $$ \sum_{n=k+1}^\infty f_n(x) = \frac{x^2}{(1+x^2)^{k}} s(x) = \frac{x^2}{(1+x^2)^k} $$ Consider, in particular, whether $0 \in [a,b]$.