What is the interval of convergence of $sum_1^oo ( (x+2)^n)/n^n$ ?

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Answer 1 · 2017-11-22T19:32:34

See below.

Using the Stirling asymptotic approximation

$n! approx sqrt(2pin)(n/e)^n$ we have for large $n$ values

$(x+2)^n/n^n approx sqrt(2pi n) ((x+2)/e)^n/(n!) = sqrt(2pi)((x+2)/e)((x+2)/e)^(n-1)/((n-1)! n^(1/2))$ or

$(x+2)^n/n^n le sqrt(2pi)((x+2)/e)((x+2)/e)^(n-1)/((n-1)!)$ and then asymptotically

$sum_(n=1)^oo (x+2)^n/n^n le sqrt(2pi)((x+2)/e)e^((x+2)/e)$ which is convergent for all $x in RR$

What is the interval of convergence of sum_1^oo ( (x+2)^n)/n^n sum_1^oo ( (x+2)^n)/n^n ?