Tuesday, 31 May 2016

calculus - Finding limxtoinftyfrac(x!)frac1xx






Find lim




I have no idea how to solve it, I can approximate it to be in (0,1) by squeezing but getting to the solution (\frac 1 e) seems like it would require a lot more. Is this an identity?




Note: no integrals nor gamma function.


Answer



Note this
\left( \frac{x!}{x^x} \right)^{1/x} = (a_x)^{1/x}



where a_x = \frac{x!}{x^x} and then use the fact that




\lim_{x\to \infty} (a_x)^{1/x} = \lim_{x\to \infty} \frac{a_{x+1}}{a_x}





and the evaluation of limit will become easy



\lim_{x\to \infty} \frac{a_{x+1}}{a_x} = \lim_{x\to \infty} \frac{1}{(1+1/x)^x} = \frac{1}{e}.


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