What is the limit of the sequence (Xn) = (n!)^(1/n)?

[steviet]

I'm working on the limit of the sequence
(Xn) = (n!)^(1/n)
Pretty sure it diverges as n goes to infinity,
but unsure how to show it.
Any hint would ge greatly appreciated.

 

[Dick]

Your instincts are good. Take a log and use Stirling's formula. That's one way.

 

[DavidSmith]

an elemetary proof is to take log so you get (1/n)log[n]! then you get [log1+Log2+log3...logn]/n

set bounds for 1<n<11 sum truncate terms
you get sum between 0 and 1

for 11<n<101 get sum between 10 and 20

you can see how this diverges