Lim n to infinity for factorial

[noobiez]

Homework Statement

lim n -> infinity for (n!)^(1/n)

Homework Equations

The Attempt at a Solution

hmm, i know that lim n approaches infinity, (n)^(1/n) will go to 1, but issit the same for n!?

 

[HallsofIvy]

You might do this: n!= 1*2*3*...* (n-1)*n so it has exactly n factors. (n!)^(1/n)= (1)^(1/n)(2)^(1/n)(3)^(1/2)*...*(n-1)^(1/n)*n^(1/n). Now you say that you know that n^(1/n) goes to 1. What do you think the other numbers go to? In particular, what does 2^(1/n) or 3^(1/n) go to? If you don't know try looking at 2^(1/100000) or 3^(1/100000). What does the product of thing like that go to?

 

[Dick]

You might do this: n!= 1*2*3*...* (n-1)*n so it has exactly n factors. (n!)^(1/n)= (1)^(1/n)(2)^(1/n)(3)^(1/2)*...*(n-1)^(1/n)*n^(1/n). Now you say that you know that n^(1/n) goes to 1. What do you think the other numbers go to? In particular, what does 2^(1/n) or 3^(1/n) go to? If you don't know try looking at 2^(1/100000) or 3^(1/100000). What does the product of thing like that go to?

?? All of those limits may be one. But the limit of the product certainly isn't one. Use Stirling's formula for n!.