I want to use Leibniz test to check uniformly convergence $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}x^2}{(1+x^2)^n}$. In order to do that I need to check whether the series $\frac{(-1)^{n-1}x^2}{(1+x^2)^n}$ uniformly converges to 0. It's easy to check that it is pintwiseconverges to 0, I'd like your help in deciding if the convergence is uniform. I thought of checking the $\lim_{n \to \infty} |f_n(x)-f(x)|$, I tried to derive $r_n(x)=f_n(x)-f(x)$, I got that $x_1=o, x_2=\frac{1}{n-1}$, both donate that the lim $r_n(x)$ is 0 when $n \to \infty$, Can I use that to conclude that the function does uniformly converges?
the radius of $x$ is all the real line.
Thanks a lot!