Series Convergence/Divergence

[The Lion 102]

Check whether the following series converges or diverges using the comparison test:

[math] \sum \dfrac{1}{n^{ln(n)}}[/math]
I'm having trouble finding the inequality that allows me to use the comparison test.

 

[tkhunny]

"Having trouble" Okay, what sort of trouble? What comparisons have you made that didn't seem to work out for you?

What do you know of 1/n and 1/n^2?

 

[pka]

Check whether the following series converges or diverges using the comparison test:
[math] \sum \dfrac{1}{n^{ln(n)}}[/math]I'm having trouble finding the inequality that allows me to use the comparison test.

Do you know that \(\sum\limits_{n = 2}^\infty {\dfrac{1}{{{n^2}}}} \) converges? SEE HERE AND HERE

 

[The Lion 102]

"Having trouble" Okay, what sort of trouble? What comparisons have you made that didn't seem to work out for you?

What do you know of 1/n and 1/n^2?

Oh I see, we can use the fact that n^ln(n)>n^2 for n>8.
1/n^2>1/n^ln(n)
The left hand side is a p-series with p>1 so it converges. Therefore, we deduce that the sum converges.
Thanks!