# Does the limit converge or diverge

## 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.

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Mark44
Mentor

## 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.

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

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

LCKurtz
Homework Helper
Gold Member
Have you thought about the integral test?

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 cant figure out what to do

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

LCKurtz
$$n^{\frac 3 2}$$