Series integral test

deficiency

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Nov 18, 2013
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Hello everyone

I was required to use integral test to find out if the next series converges or diverges, but I'm not sure how to start.
Hope you can help me, thanks.

Series:

∑ 1/ n ln n ln (ln n)
n = 3
 
Series:

∑ 1/ n ln n ln (ln n)
n = 3
Does the above mean either of the following?

. . . . .\(\displaystyle \mbox{a) }\, \displaystyle{\sum_{n=3}^{\infty}}\, \left(\dfrac{1}{n}\right)\ln(n) \ln(\ln(n))\)

. . . . .\(\displaystyle \mbox{b) }\, \displaystyle{\sum_{n=3}^{\infty}}\, \dfrac{1}{n \ln(n) \ln(\ln(n))}\)

Or something else?

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you! ;-)
 
Does the above mean either of the following?

. . . . .\(\displaystyle \mbox{a) }\, \displaystyle{\sum_{n=3}^{\infty}}\, \left(\dfrac{1}{n}\right)\ln(n) \ln(\ln(n))\)

. . . . .\(\displaystyle \mbox{b) }\, \displaystyle{\sum_{n=3}^{\infty}}\, \dfrac{1}{n \ln(n) \ln(\ln(n))}\)

Or something else?

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you! ;-)

Yes! I meant the second one, but I already figured it out. I can use integral test because is positive, continuous and decreasing. So I take the limit of that and at the end I will have something like

Lim ln(ln(lnx)) | = ln (ln (lnb)) - ln (ln (ln3)) = infinity .. so it diverges.
b -> infinity

Am I right?
 
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