Evaluate Let $f_{n}(x) = \frac{x^{2}}{(1+x^{2})^{n}}$ for $x \in \mathbb{R}$. Evaluate the sum $$S(x) = \sum_{n = 0}^{\infty}f_{n}(x)$$.
I'm really at a lost of how to approach this. It isn't asking me to show convergence and if that was the case I have no idea to what it converges to. If I did know what it converges to then perhaps I could deduce what the sum would be. This question is in a chapter about applying the M-Test for series of functions and uniform convergence. What would be some ways to evaluate this?
If $x\neq0$, then, since$$\sum_{n=0}^\infty\frac1{(1+x^2)^n}=\sum_{n=0}^\infty\left(\frac1{1+x^2}\right)^n=\frac1{1-\frac1{1+x^2}}=\frac{1+x^2}{x^2},$$we have$$\sum_{n=0}^\infty\frac{x^2}{(1+x^2)^n}=1+x^2.$$If $x=0$, then the sum is $0$, of course.