Home > Calculus > Determining the Radius and Interval of Convergence for a Power Series

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

Answer 1

#(-1/2,1/2)#

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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Answer 2

Aurora Alvarado

The interval of convergence for the series ( \sum{(-1)^n 2^n x^n} ) is ((-1/2, 1/2)).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.