This problem is from a calculus competition.

What is the radius of convergence of $\sum_{n=1}^{\infty} \frac{(xn)^n}{n!}$?

I know the answer is 1/e, but the solution uses Stirling's approximation which I am not familiar with. Is there any way to do this using the ratio test?

  • 2
    $\begingroup$ How about the ratio test? $\endgroup$ – zhw. Jun 28 '17 at 1:43

There is:

\begin{align} \frac{(n + 1)^{n + 1}/(n + 1)!}{n^n/n!} &= \frac{(n + 1)^{n}(n+1)/(n + 1)!}{n^n/n!} \\ &= \frac{(n + 1)^{n}/n!}{n^n/n!} \\ &= \frac{(n + 1)^{n}}{n^n} \\ &= \left( 1 + \frac1n \right)^n \\ &\to e. \end{align}


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