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