Help with series convergence or divergence

**Cakecake** · January 12th 2009, 03:35 PM

I need help to determine if

$\frac{ln n}{n}$

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.

**Plato** · January 12th 2009, 03:42 PM

Originally Posted by Cakecake

I need help to determine if $\frac{ln n}{n}$ 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 $\sum\limits_{n = 1}^\infty {\frac{{\ln (n)}}{n}}$ ?

**Cakecake** · January 12th 2009, 03:46 PM

Ah yes, I forgot the summation.

**Mathstud28** · January 12th 2009, 03:50 PM

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

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

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