Does the limit converge or diverge

[kuczmama]

Homework Statement

infinity
Ʃ \frac{ln(n)}{n\sqrt{n}}
n=1

Homework Equations

I think that I need to use either the limit comparison test or the ratio test.

The Attempt at a Solution

After listing all the terms I found out that the function was positive and decreasing, and that the terms approached 0. However when I used the limit comparison test or the root test I keep on getting an answer of infinity or 0, which is inconclusive. So I don't know how else to approach the problem. Is there something easy that I am overlooking. Thanks.

 

[Mark44]

Homework Statement

infinity
Ʃ \frac{ln(n)}{n\sqrt{n}}
n=1

Homework Equations

I think that I need to use either the limit comparison test or the ratio test.

The Attempt at a Solution

After listing all the terms I found out that the function was positive and decreasing, and that the terms approached 0. However when I used the limit comparison test or the root test I keep on getting an answer of infinity or 0, which is inconclusive. So I don't know how else to approach the problem. Is there something easy that I am overlooking. Thanks.

Show us what you tried for the limit comparison test. I agree that the ratio test would not be conclusive.

 

[deluks917]

I think the best test to use is the ratio test:

If there exists a constant C < 1 such that |a_n+1/a_n|<C for all sufficiently large n, then ∑an converges absolutely.

 

[LCKurtz]

Have you thought about the integral test?

 

[kuczmama]

for the limit comparison test I tried a bunch of different comparisons. I said as lim n-> infinity of

lim n→∞. of (ln(n)/(nsqrt(n))/(1/n^5/4).That didnt give me any help. Then I tried comparing it to 1/n^1/4. Then I tried comparing it to a bunch of other stuff but nothing seemed to work. I can't figure out what to do

 

[kuczmama]

Also I never thought about the integral test. I will try that

 

[LCKurtz]

@kuczmama: Don't use the X² key inside tex brackets. Right click on this to see how to do exponents:

n^{\frac 3 2}

 

[kuczmama]

I got that the integral converges to 4. I guess that this means the sum also converges. Thanks a lot guys!