Find all 
such that the series
![\[ \sum_{n=0}^{\infty} nx^n \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-16fe1f7cf63ab76b4283bf17e2812c28_l3.png "Rendered by QuickLaTeX.com")
converges and compute the sum.
First, we write
![\[ \sum_{n=0}^{\infty} nx^n = x \cdot \sum_{n=0}^{\infty} nx^{n-1}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-f571746250c1d5009308ec1afd12a9e3_l3.png "Rendered by QuickLaTeX.com")
By Theorem 11.9 we know that we can differentiate a power series term-by-term, so we have
![\begin{align*} x \cdot \sum_{n=0}^{\infty} nx^{n-1} &= x \cdot D_x \left( \sum_{n=0}^{\infty} x^n \right) \\[9pt] &= x \cdot D_x \left( \frac{1}{1-x} \right) &(\text{Geometric series})\\[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-37e9f4375b2c579c4cbda5e6ac39c696_l3.png "Rendered by QuickLaTeX.com")
valid for 
(since this was where the geometric series was valid).