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Integral Test
Trying to determine whether the following series converges or diverges using the intergral test:
$\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n\ln \left( n+1 \right) }}$
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
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Can we use comparison test before using integral test ? ie $\displaystyle \frac{1}{n} > \frac{1}{n+1}$ and
$\displaystyle \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 $\displaystyle \int \frac{dx}{ x \ln{x} } = \ln{\ln{x}} + C $