# Proving Convergence of this

1. Mar 22, 2009

### simba924

1. The problem statement, all variables and given/known data

$$\sum$$ from n=2 to $$\infty$$ of n/((n2-5)*(ln n)2)

2. Relevant equations

3. The attempt at a solution
I've tried Limit Comparison but I always get a limit of 0 which will not work. Ratio test doesn't help. I don't think a direct comparison can be made but that seems to be the only other option...
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 22, 2009

### yyat

Hi simba924!

To simplify the problem a bit, show that

$$\frac{n}{n^2-5}\le\frac{2}{n}$$

for large n. Then apply the http://en.wikipedia.org/wiki/Integral_test_for_convergence" [Broken].

Last edited by a moderator: May 4, 2017
3. Mar 22, 2009

### simba924

Ahh thanks. Can you explain that in a little more detail though, I don't really get it.

4. Mar 22, 2009

### simba924

Hey I still need some help with this one. I'm pretty sure that $$\sum$$ 2/n is not convergent because it is in form integer/np where p=1

Can anyone help?

5. Mar 23, 2009

### yyat

Remember, you still have the additional factor 1/(ln n)^2 (without which it would diverge). To do the integral test you have to check that

$$\int_2^{\infty}\frac{1}{x(\log(x))^2}$$

exists (you can find an explicit antiderivative, Hint: substitution).