# How do you find the interval of convergence #Sigma 2x^n# from #n=[0,oo)#?

The radius of convergence is

We will use the ratio test for convergence. We have that:

The ratio test tells us that the sum convergence if:

So:

So, by the ratio test, for convergence:

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The interval of convergence for the series Σ 2x^n from n=0 to infinity can be found using the ratio test.

First, apply the ratio test by taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term.

The (n+1)th term of the series is 2x^(n+1), and the nth term is 2x^n.

So, the ratio is |2x^(n+1) / 2x^n| = |x|.

Now, take the limit as n approaches infinity of |x|, which is simply |x|.

For the series to converge, |x| must be less than 1. Therefore, the interval of convergence is -1 < x < 1.

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