Monday, April 29, 2019

calculus - Convergence of sumin=1nftyfracn!,innn



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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