Integral Test

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May 18th 2010, 10:06 PM
Mathman87

Integral Test

Trying to determine whether the following series converges or diverges using the intergral test:

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

I know the theory of how to use the integral test but im struggling to get this into an easily integratable form. I tried using substitution like y=ln(n+1) etc. but could get anywhere. Any ideas?

Thanks
May 18th 2010, 10:16 PM
simplependulum

Can we use comparison test before using integral test ? ie $\frac{1}{n} > \frac{1}{n+1}$ and

$\sum_{n=1}^{\infty} \frac{1}{n \ln(n+1)} >\sum_{n=1}^{\infty} \frac{1}{(n+1)\ln(n+1)} = \sum_{n=2}^{\infty} \frac{1}{n \ln(n)}$

and we have $\int \frac{dx}{ x \ln{x} } = \ln{\ln{x}} + C$

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