Answer
By Theorem 9.6, the series converges.
Work Step by Step
Th.9.6, Convergence of a Geometric Series:
A geometric series with ratio $r$ diverges when $|r| \geq 1$.
If $0 < |r| < 1$,
then the series converges to the sum $\displaystyle \sum_{n=0}^{\infty}ar^{n}=\frac{a}{1-r},\quad 0 < |r| < 1$.
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$\displaystyle \sum_{n=0}^{\infty}2\left(-\frac{1}{2}\right)^{n}$ is a geometric series with
$|r|=\left|-\displaystyle \frac{1}{2}\right| < 1$
By Theorem 9.6, the series converges.
