The root test states that :

(1) If

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then the series converges absolutely.

(2) If

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then the series diverges.

(3) If

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and the limit approaches strictly from above then the series diverges.

(4) Otherwise the test is inconclusive (the series may diverge, converge absolutely or converge conditionally).

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Apply root test on

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Substitute

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The condition is

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for convergence.

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Therefore the value is

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Therefore the radius of convergence of the power series

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The radius of convergence of the power series

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