sequences and series - Finding the convergence radius of $sum_{n=1}^{infty}frac{n!}{n^n}(x+2)^n$

I wish to find the convergence radius of the power series:

$$\sum_{n=1}^{\infty}\frac{n!}{n^n}(x+2)^n$$

I found the radius is $-e-2\lt x \lt e-2$ and this was also the answer in the book.

My problem is showing the power series does not converge at the edges.

For $x=e-2$ I used the ratio test and got a limit bigger than one.

For $x=-e-2$ I get an alternating series. how do I show it does not converge?

Answer

A useful inequality:

$$n!>\left(\frac ne\right)^n$$

which should help you show that it fails the term test at the boundaries.

The derivation of this follows by taking the log of both sides:

$$\ln(n!)>n\ln(n)-n$$

By the definition of the factorial, some log rules, and a Riemann sum,

$$\ln(n!)=\sum_{k=1}^n\ln(k)=\int_0^n\ln\lceil x\rceil~\mathrm dx>\int_0^n\ln(x)~\mathrm dx=n\ln(n)-n$$