I have the following progression $$ \frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + ... + \frac{1}{(1+x^2)^n} $$ I have that $a=\frac{1}{1+x^2}$ and $q=\frac{1}{1+x^2}$, then using $a\frac{1-q^{n+1}}{1-q}$ I got $\frac{(1+x^2)^{n+1} - 1}{x^2(1+x^2)^{n+1}}$
My solution seems to be a bit hard and I think that it's possible to simplify the solution above. I will be grateful for any help you can provide.