What is the interval of convergence of $sum [(-1)^n(2^n)(x^n)]$ ?

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What is the interval of convergence of $sum [(-1)^n(2^n)(x^n)]$ ?

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Answer 1 · 2016-03-21T17:46:30

$(-1/2,1/2)$

Explanation:

First, note that $sum[(-1)^n2^nx^n] = sum(-2x)^n$

A geometric series of the form $sumr^n$ converges if and only if $|r| < 1$ . Then, for the given series to converge, we must have
$|-2x| < 1$ , that is, $|x| < 1/2$ . Thus the interval of convergence for the series is $(-1/2,1/2)$

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What is the interval of convergence of $sum [(-1)^n(2^n)(x^n)]$ ?

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What is the interval of convergence of sum [(-1)^n(2^n)(x^n)] sum [(-1)^n(2^n)(x^n)] ?

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What is the interval of convergence of $sum [(-1)^n(2^n)(x^n)]$ ?