Thursday, October 24, 2019

calculus - Find limntoinftyfrac(n!)1/nn.






Find lim




I don't know how to start. Hints are also appreciated.


Answer




let y= \frac{(n!)^{1/n}}{n}.
\implies y=\left (\frac{n(n-1)(n-2)....3.2.1}{n.n.n....nnn}\right)^\frac{1}{n}
Note that we can distribute the n in the denominator and give an 'n' to each term
\implies \log y= \frac {1}{n}\left(\log\frac{1}{n}+\log\frac{2}{n}+...+\log\frac{n}{n}\right)
applying \lim_{n\to\infty} on both sides, we find that R.H.S is of the form



\lim_{n\to\infty} \frac{1}{n} \sum_{r=0}^{r=n}f \left(\frac{r}{n}\right)
which can be evaluated by integration =\int_0^1\log(x)dx
=x\log x-x plugging in the limits (carefully here) we get -1.




\log y=-1,\implies y=\frac{1}{e}


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