Integral Test

Apr 2010
30
0
Hi, im trying to perform an integral test on the following:

\(\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }}\)

I know the theory is to integrate the function and then to evaluate it and see if it is finite... however i am having some problems with the integration. Can anyone help?
 

skeeter

MHF Helper
Jun 2008
16,216
6,764
North Texas
Hi, im trying to perform an integral test on the following:

\(\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }}\)

I know the theory is to integrate the function and then to evaluate it and see if it is finite... however i am having some problems with the integration. Can anyone help?
use the integral test to show \(\displaystyle \sum_{n=2}^\infty \frac{1}{n\ln{n}}\) diverges ... then use the limit comparison test with this divergent series to show \(\displaystyle \sum_{n=1}^\infty \frac{1}{n\ln(n+1)}\) also diverges.
 
Mar 2010
107
14
\(\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }} \) behaves like \(\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n \right) }} \); thus, \(\displaystyle \int_1^\infty \frac{1}{n\ln(n)} dn = \left(\ln \ln(n)\right) |_1^\infty \)