Find all 
such that the series
![\[ \sum_{n=0}^{\infty} (-1)^n x^{2n} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1e95272c32165cb41906c6ec78d08c17_l3.png "Rendered by QuickLaTeX.com")
converges and compute the sum.
We have
![\[ \sum_{n=0}^{\infty} (-1)^n x^{2n} = \sum_{n=0}^{\infty} (-x^2)^n = \frac{1}{1-(-x^2)} = \frac{1}{1+x^2} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-0eae3caf962ec8a192b37abd92a00706_l3.png "Rendered by QuickLaTeX.com")
by the expansion for the geometric series. This is convergent for 
.
Find all such that the series
converges and compute the sum.
We have
by the expansion for the geometric series. This is convergent for .