Help with condensation test

i guys.

i need to determine if this serie converges:

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

this is what i've tried so far:

i use condensation test and got the following:

$\displaystyle \sum \frac{2^n}{2^n\ln2^n[\ln(\ln2^n)]}=\frac{1}{n\ln2[\ln(n\ln2)]}$

$\displaystyle \frac{1}{\ln2}\sum \frac{1}{n[\ln(n\ln2)]}=\frac{1}{n[\ln n+\ln(\ln2)]}=\frac{1}{n\ln n+n\ln(\ln2)}$

$\displaystyle \frac{1}{\ln2}\sum \frac{1}{n\ln n+n\ln(\ln2)}=\frac{1}{n\ln n}+\frac{1}{n\ln(\ln2)}$

now, my question is this:

can i say that since: $\displaystyle \sum \frac{1}{n\ln(\ln2)}$ does not converge the the whole thing does'nt converge?

is there more elegant way to handle this horrible serie?

thanks in advanced!

Re: Help with condensation test

Hey Stormey.

Have you come across the Integral Test for convergence? I would suggest looking into that with the appropriate substitution.

Re: Help with condensation test

Yes, chiro is right, the integral test works.

- Hollywood

Here is a link to the condensation test: Cauchy condensation test - Wikipedia, the free encyclopedia

Re: Help with condensation test

hi, and thanks for the help.

we didn't learn the integral test yet, so i'm supposed to answer it without it, i guess...

Re: Help with condensation test

Maybe this condensation test will do it.

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

which converges if and only if the following series converges:

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

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

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

This is what you did in your original post. Now let's do the condensation test again. Our original series converges if and only if the following series converges:

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

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

We think this diverges, so we simplify by making it smaller. We have $\displaystyle \ln(ln(2))\le 2\ln(2)\le n\ln(2)$, so our new sum is greater than or equal to:

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

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

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

which diverges.

- Hollywood

Re: Help with condensation test

brilliant!

thank you very much Hollywood.

didn't know i can use this test twice like that. :)