Find the positive values of $\displaystyle p$ for which the series converges:

$\displaystyle \sum_{n=2}^{\infty} \frac {1}{n (\ln n)^p}$

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- Apr 6th 2008, 10:43 AMdark_knight_307Convergence with multiple variables
Find the positive values of $\displaystyle p$ for which the series converges:

$\displaystyle \sum_{n=2}^{\infty} \frac {1}{n (\ln n)^p}$ - Apr 6th 2008, 10:53 AMMoo
Hello,

Take a look at this : Bertrand Series ;) - Apr 6th 2008, 10:56 AMMathstud28OK go through with the integral test
$\displaystyle \int_1^{\infty}\frac{1}{n\ln(n)^{p}}dn$=$\displaystyle \frac{(\ln(n))^{-p+1}}{-p+1}+C$ and I think you can see where to go from there...just remmber that it must converge(the integral) for the series to converge..oh yeah and if $\displaystyle p=1$ then the antiderivative is $\displaystyle \ln(\ln(n))$ +C

- Apr 6th 2008, 11:05 AMdark_knight_307
But you did it from 1 to infinity, and it is 2 to infinity.......

- Apr 6th 2008, 11:12 AMKrizalid
Do it when $\displaystyle n=2$ and apply the integral test.

- Apr 6th 2008, 11:12 AMdark_knight_307
Lemme check my answer, is it when p > 1?

- Apr 6th 2008, 11:13 AMMathstud28Yes
I think thats right