Infinite Series Convergence Test: ln((n!e^n)/n^(n+1/2)) [SOLVED]

[garryh22]

[SOLVED] Infinite Series

Homework Statement

ln((n!e^n)/n^(n+1/2))

Homework Equations

Does the series above converge or diverge.

The Attempt at a Solution

I can see that it diverges but I'm looking for the appropriate test to show this

 

[Roberto Bramb]

The series is convergent by Cauchy criterium since C(n+1)/C(n)->0 for n>infinity,
as can be easily verified using Striling formula : log(n!) => n*log(n)-n

 

[garryh22]

Thanks a great deal. I never heard of the Striling formula till now. I just looked it up, applied it and the expression reduced to ln(1/n)^(1/2) which diverges with p-series. Further insight would be greatly appreciated