# Integral Test

#### 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?

#### skeeter

MHF Helper
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.

#### 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, $$\displaystyle \int_1^\infty \frac{1}{n\ln(n)} dn = \left(\ln \ln(n)\right) |_1^\infty$$