Convergence with multiple variables

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April 6th 2008, 10:43 AM
dark_knight_307

Convergence with multiple variables

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

$\sum_{n=2}^{\infty} \frac {1}{n (\ln n)^p}$
April 6th 2008, 10:53 AM
Moo

Hello,

Take a look at this : Bertrand Series ;)
April 6th 2008, 10:56 AM
Mathstud28

OK go through with the integral test

$\int_1^{\infty}\frac{1}{n\ln(n)^{p}}dn$ = $\frac{(\ln(n))^{-p+1}}{-p+1}+C$ 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 $p=1$ then the antiderivative is $\ln(\ln(n))$ +C
April 6th 2008, 11:05 AM
dark_knight_307

But you did it from 1 to infinity, and it is 2 to infinity.......
April 6th 2008, 11:12 AM
Krizalid

Do it when $n=2$ and apply the integral test.
April 6th 2008, 11:12 AM
dark_knight_307

Lemme check my answer, is it when p > 1?
April 6th 2008, 11:13 AM
Mathstud28

Yes

I think thats right

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