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 $|x| < 2$ ).

Stumbling Robot

Verify that 1 / (2-x) = ∑ xⁿ / 2ⁿ⁺¹

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