Home » Blog » Without justification, establish the given formula

Without justification, establish the given formula

Obtain the following formula without attempting to justify the steps used in the process.

    \[ \sum_{n=1}^{\infty} n^4x^n = \frac{x^4 + 11x^3 + 11x^2 + x}{(1-x)^5}. \]


Starting with the formula

    \[ \sum_{n=1}^{\infty} n^3 x^n = \frac{x^3 + 4x^2 + x}{(1-x)^4} \]

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*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):