Calculate a limit with n!

[mosh]

I need to prove that
http://www.wolframalpha.com/input/?i=lim+n->inf+3^2^n/(n!)^n

I tried many things, including Stiriling's formula for n! but no luck...

 

[tkhunny]

I need to prove that
http://www.wolframalpha.com/input/?i=lim+n->inf+3^2^n/(n!)^n

I tried many things, including Stiriling's formula for n! but no luck...

See if a logarithm transformation helps. \(\displaystyle \dfrac{2^{n}\cdot\log(3)}{n\cdot\log(n!)}\)

 

[mosh]

I get in the denominator nlog(1)+nlog(2)+...+nlog(n) but I'm stuck there. One more hint please?