Question

Does the following series converge or diverge?

Answer 1

\[\sum_{n=2}^{infinity} 1/n \sqrt{\ln(n)}\]

Answer 2

I'm wanting to say it converges to 1, but I'm not sure how to prove that. Real analysis wasn't my strong point.

Answer 3

do you know what test to use?

Answer 4

my original thought was ratio, but i dont know how to apply that here

Answer 5

I'm wanting to say it has something to do with the Cauchy Criterion

Answer 6

...I have no idea what that means..haha

Answer 7

I don't think we covered that

Answer 8

Oh, the ratio test should work.

Answer 9

really? Let me try that really quickly

Answer 10

Well, that gave me infinity over infinity...maybe i'm doing it wrong?

Answer 11

Looking up properties of the natural log right now, give me a sec.

Answer 12

thanks so much!

Answer 13

Integral test would be easier since i see ln(n) and 1/n there.

Answer 14

oh that makes sense!

Answer 15

Ok all i can think of is that \[n / (n+1) \le 1\]
and
\[\sqrt(\ln(n+1)) \le n\]
and \[\sqrt(\ln(n)) \le n\]
thus
\[\left| n/(n+1) * \sqrt(\ln(n+1)) / \sqrt(\ln(n)) \right| \le \left| 1 * n/n \right| \le 1\]

Answer 16

hmm. Ok, let me think about that for a min :)

Answer 17

hmm actually that might not work cause the ration needs to be less than 1 for the ratio test to work

Answer 18

when i worked it doing the integral test i got infinity meaning divergent does that seem right?