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Without justification, establish the given formula

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

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


From the previous exercise (Section 10.7, Exercise #17) we have the formula

    \[ \sum_{n=0}^{\infty} (n+1)x^n = \frac{1}{(1-x)^2}. \]

Differentiating both sides we then have

    \begin{align*}  && \sum_{n=0}^{\infty} (n+1)x^n &= \frac{1}{(1-x)^2}\\[9pt]  \implies && \sum_{n=0}^{\infty} n(n+1)x^{n-1} &= \frac{2}{(1-x)^3} \\[9pt]  \implies && \sum_{n=-1}^{\infty} (n+1)(n+2) x^n &= \frac{2}{(1-x)^3} \\[9pt]  \implies && \frac{1}{2} \sum_{n=0}^{\infty} (n+1)(n+2)x^n &= \frac{1}{(1-x)^3} \\[9pt]  \implies && \sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{2!}x^n &= \frac{1}{(1-x)^3}.  \end{align*}

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