real analysis - Find the radius of convergence for the series $sum_{k=0}^{infty}frac{k!}{k^k}x^k$.

Find the radius of convergence for the series $\sum_{k=0}^{\infty}\frac{k!}{k^k}x^k$.

For other similar problems, I could apply the Ratio Test or the Root Test to find the radius of convergence. For this problem, these tests are not seem to be working. The book says I should take reference to the power series of $e^x$ to determine the endpoints but I can't even find the endpoints of the radius of convergence.

Answer

Use Hadamard's formula: the radius of convergence $R$ is given by
$$\frac1R=\limsup a_k^{1/k}.$$

In the present case, Stirling's formula gives the answer:

$$\biggl(\frac{k!}{k^k}\biggr)^{\!1/k}\sim_\infty\left(\frac{\sqrt{2\pi k}\Bigl(\dfrac ke\Bigr)^k}{k^k}\right)^{\!1/k}=\frac{(2\pi k)^{1/2k}}e\to\frac1e.$$