Obtain the following formula without attempting to justify the steps used in the process.
![\[ \sum_{n=1}^{\infty} n^2 x^n. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-4ee61096170551b8169f6088b642e5fd_l3.png "Rendered by QuickLaTeX.com")
We follow a similar procedure to the previous exercise (Section 10.7, Exercise #11), starting with the formula we derived in that exercise,
![\begin{align*} && \sum_{n=1}^{\infty} nx^n &= \frac{x}{(1-x)^2} \\[9pt] \implies && \sum_{n=1}^{\infty} n^2 x^{n-1} &= \frac{1+x}{(1-x)^3} &(\text{differentiating}) \\[9pt] \implies && \sum_{n=1}^{\infty} n^2 x^n &= \frac{x+x^2}{(1-x)^3} &(\text{multiply by }x). \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a7f7f5bb98af5effdc46d08ee608b5dc_l3.png "Rendered by QuickLaTeX.com")
Obtain the following formula without attempting to justify the steps used in the process.
We follow a similar procedure to the previous exercise (Section 10.7, Exercise #11), starting with the formula we derived in that exercise,