Find the radius and interval of convergence of ∑ 0,∞ n^{n}x^{n}/n!
Find the radius and interval of convergence
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1 Answer
Find the ratio of successive terms:
|a_{n+1}/a_{n} | = | (n+1)^{n+1}/n^{n} * n!/(n+1)! * x| = |(n+1)^{n}/n^{n} 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}