Question

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

Answer 1

See below.

Explanation:

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`