I need to study this sum: ∑∞n=1n!innn. Taking: lim
\rightarrow \lim_{n\rightarrow \infty} \left|\left(\frac{n!}{n^n}\right)^{\frac{1}{2}}\right|
Using Stirling approximation considering the limit:
\ln n! \approx n \ln\:n-n
then n! \approx (\frac{n}{e})^n (is this correct?):
\rightarrow \lim_{n\rightarrow \infty} \left|\frac{n}{n\, e}\right| = \frac{1}{e} This doesn't make too much sense because I know the series diverges.
I think I'm missing a \sqrt{2\pi n} in Stirling approximation but I don't understand why that pops out taking the e^{(\;)} from the first expression.
Answer
The series is absolutely convergent. You don't need Striling's approximation. Just apply ratio test: \frac {|a_{n+1}|} {|a_n|} \to \frac 1 e.
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