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Find the radius and interval of convergence

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1 Answer

 
Find the ratio of successive terms:
|an+1/an | = | (n+1)n+1/nn * n!/(n+1)! * x| = |(n+1)n/nn x| = |(1+1/n)n x| → e* |x|
For convergence,
|x| < e-1
The radius of convergence is e-1.
The interval of convergence is at least
|x| < e-1, or  -e-1 < x < e-1.
The convergence at the endpoints x=±e-1 is not obvious: the ratio test fails here. According to Wolfram Alpha, the series converges at x=-e-1 but not at x=e-1, so the interval is
-e-1 ≤ x < e-1