# Thread: Help with series convergence or divergence

1. ## Help with series convergence or divergence

I need help to determine if

$\displaystyle \frac{ln n}{n}$

converges or diverges. By the nth term test it equals 0 so it could be either. I remember doing this in class but I forgot.

2. Originally Posted by Cakecake
I need help to determine if $\displaystyle \frac{ln n}{n}$ converges or diverges. By the nth term test it equals 0 so it could be either. I remember doing this in class but I forgot.
Do you mean $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{\ln (n)}}{n}}$?

3. Ah yes, I forgot the summation.

4. To determine the convergence of $\displaystyle \sum_{n=1}^{\infty}\frac{\ln(n)}{n}$ you can do a couple things.

1) Use the integral test which states that if $\displaystyle a_n$ is continuous, nonincreasing, and positive that $\displaystyle \sum_{n=c}^{\infty} a_n$ and $\displaystyle \int_c^{\infty} f~dx$ share convergence

2) Use Cauchy's condensation test. Once again if the same hypotheses apply then $\displaystyle \sum a_n$ and $\displaystyle \sum a_{2^n}\cdot 2^n$ share convergence.

5. Originally Posted by Plato
Do you mean $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{\ln (n)}}{n}}$?
Note that $\displaystyle \frac{\ln (n)}{n} > \frac{1}{n}$ for $\displaystyle n > 2$.

So use the comparison test to prove divergence ....

6. Ooo right thanks, that's what I did over in class. Didn't bother thinking of it because I was using that to compare another series.