# Integral Test

• May 10th 2010, 04:22 PM
Mathman87
Integral Test
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?
• May 10th 2010, 05:01 PM
skeeter
Quote:

Originally Posted by Mathman87
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.
• May 10th 2010, 05:09 PM
lilaziz1
$\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }}$ behaves like $\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n \right) }}$; thus, $\displaystyle \int_1^\infty \frac{1}{n\ln(n)} dn = \left(\ln \ln(n)\right) |_1^\infty$