Find all 
such that the series
![\[ \sum_{n=0}^{\infty} (-1)^n nx^n \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-27699f85e1f5726a380cecaa1bf3affc_l3.png "Rendered by QuickLaTeX.com")
converges and compute the sum.
First, we write
![\[ \sum_{n=0}^{\infty} (-1)^n n x^n = x \cdot \sum_{n=0}^{\infty} (-1)^n nx^{n-1}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-c136c14a28756678e95236b463e1fc8b_l3.png "Rendered by QuickLaTeX.com")
Then, from Theorem 11.9 we know we can differentiate term-by-term and we have
![\begin{align*} x \cdot \sum_{n=0}^{\infty} (-1)^n nx^{n-1} &= x \cdot D_x \left( \sum_{n=0}^{\infty} (-1)^n x^n \right) \\[9pt] &= x \cdot D_x \left( \sum_{n=0}^{\infty} (-x)^n \right) \\[9pt] &= x \cdot D_x \left( \frac{1}{1+x} \right) &(|x| < 1)\\[9pt] &= x \cdot \left( \frac{-1}{(1+x)^2} \right)\\[9pt] &= \frac{-x}{(1+x)^2}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-88be1069de209bb4a990cb3ffbd60b1e_l3.png "Rendered by QuickLaTeX.com")
Find all such that the series
converges and compute the sum.
First, we write
Then, from Theorem 11.9 we know we can differentiate term-by-term and we have