1. ## Uniform convergence....

Prove that $\Sigma_{n=1}^\infty \frac{x^2}{(x^2+1)^n}$ uniformly convergence at $[a,\infty)$ for $a>0$, and not uniformly convergence at $[-a,a]$ for $a>0$.

Thank you so much!

2. There isn't any $n$ in the expression...

3. To prove uniform convergence on $[a, \infty)$ you could use the Weierstrass M-test - Wikipedia, the free encyclopedia.

You can use the estimation
$\displaystyle\frac{x^2}{(x^2+1)^n}\leq \frac{x^2}{{n\choose 2} x^4}=\frac{1}{x^2} \frac{1}{{n\choose 2}}\leq \frac{1}{a^2} \frac{1}{{n\choose 2}}$ for n>1

Hope that helps.