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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