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Without justification, establish the given formula

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

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


Starting with the formula for the sum of a geometric series and differentiating both sides (without justifying that we can do this) we have

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

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