Given $$\sum_{n=0}^\infty \frac{x^2}{(1+x^2)^n}$$
I'm trying to find the nth partial sum, in order to test for convergence of the series. How would I go about doing this?
I know there are other ways of going about it but I'm trying to learn this particular one.
Best Answer
Hint 1: If $r\neq 1$ we have the formula $$ \sum_{n=0}^{k-1}a\cdot r^n = a\frac{1-r^k}{1-r} $$ and if $r=1$ we have the formula $$ \sum_{n=0}^{k-1}a\cdot r^n=a\cdot k $$ Can you prove these two formulas? Subhint: Google "geometric series".
Hint 2: The partial sums of your series can be written as $$ \sum_{n=0}^{k-1}\frac{x^2}{(1+x^2)^n}=\sum_{n=0}^{k-1}x^2\cdot\left(\frac{1}{1+x^2}\right)^n $$
Can you combine these two hints to obtain a formula?