Integral Test

**Mathman87** · May 10th 2010, 04:22 PM

Hi, im trying to perform an integral test on the following:

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

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** · May 10th 2010, 05:01 PM

Originally Posted by Mathman87

Hi, im trying to perform an integral test on the following:

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

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 $\sum_{n=2}^\infty \frac{1}{n\ln{n}}$ diverges ... then use the limit comparison test with this divergent series to show $\sum_{n=1}^\infty \frac{1}{n\ln(n+1)}$ also diverges.

**lilaziz1** · May 10th 2010, 05:09 PM

$\sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }}$ behaves like $\sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n \right) }}$ ; thus, $\int_1^\infty \frac{1}{n\ln(n)} dn = \left(\ln \ln(n)\right) |_1^\infty$

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