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Verify that 1 / (2-x) = ∑ xn / 2n+1

Assuming that \frac{1}{2-x} has a power-series representation in terms of powers of x, verify that it has the form

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

valid for real x with |x| < 2.

First, we write

    \[ \frac{1}{2-x} = \frac{1}{2} \left( \frac{1}{1 - \left( \frac{x}{2} \right)} \right). \]

Then, we recognize this the formula for a geometric series and we have

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

This is valid for |x| < 2 since that is where the geometric series expansion is valid (since the geometric series is in \frac{x}{2}, and this is valid for \left|\frac{x}{2}\right| < 1 or |x| < 2).

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