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

garryh22
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[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
 
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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
 
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
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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