Convergence and Divergence of Riemann Sum Series with Varying p

[413]

For what values of p>0 does the series

Riemann Sum [n=1 to infinity] 1/ [n(ln n) (ln(ln n))^p]

converge and for what values does it diverge?

How do i do this question? Would somebody please kindly show me the steps? Do i use the intergral test?

 

[StatusX]

Try bounding it by the series 1/n^(1+e), e>0. Note these sums get arbitrarily large as e->0.

 

[413]

it is actually a example question, my instructor said that it is divergent for all p, but i don't get it because actually i didn't copy all of the notes. Do you mind explaining it to me?

 

[StatusX]

Oh, actually the integral test does work. You can do it nicely with a substitution. Also, be careful with the first two terms. ln(ln(1))=-infinity and ln(1)=0, so the first term isn't well defined, but even if you take the limit it blows up. And ln(ln(2)) is negative, and so can't be raised to the pth power in a well defined way. I would assume the series is supposed to start at n=3.