Thursday, October 24, 2019

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






Find limn(n!)1/nn.




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


Answer




let y=(n!)1/nn.


y=(n(n1)(n2)....3.2.1n.n.n....nnn)1n

Note that we can distribute the n in the denominator and give an n to each term
logy=1n(log1n+log2n+...+lognn)

applying limn on both sides, we find that R.H.S is of the form



limn1nr=nr=0f(rn)


which can be evaluated by integration =10log(x)dx

=xlogxx
plugging in the limits (carefully here) we get 1.




logy=1,y=1e


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