Obtain the following formula without attempting to justify the steps used in the process.
![\[ \sum_{n=1}^{\infty} n^4x^n = \frac{x^4 + 11x^3 + 11x^2 + x}{(1-x)^5}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-25bd3059b938f91a21083d8b9a95c8f7_l3.png "Rendered by QuickLaTeX.com")
Starting with the formula
![\[ \sum_{n=1}^{\infty} n^3 x^n = \frac{x^3 + 4x^2 + x}{(1-x)^4} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-3d1882a0384e261f972f6e4a28b5e3bb_l3.png "Rendered by QuickLaTeX.com")
which we derived in the previous exercise (Section 10.7, Exercise #13) we have
![\begin{align*} && \sum_{k=1}^{\infty} n^3 x^n &= \frac{x^3 + 4x^2 + x}{(1-x)^4} \\[9pt] \implies && \sum_{k=1}^{\infty} n^4 x^{n-1} &= \frac{x^3 + 11x^2 + 11x + 1}{(1-x)^5} &(\text{differentiating})\\[9pt] \implies && \sum_{k=1}^{\infty} n^4 x^n &= \frac{x^4 + 11x^3 + 11x^2 + x}{(1-x)^5} &(\text{multiplying by } x). \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-3dbc9a13c8cfcd7655805b249208d56b_l3.png "Rendered by QuickLaTeX.com")
Obtain the following formula without attempting to justify the steps used in the process.
Starting with the formula
which we derived in the previous exercise (Section 10.7, Exercise #13) we have