# How do you test the series Sigma lnn/n from n is [1,oo) for convergence?

Mar 9, 2017

The series:

${\sum}_{n = 1}^{\infty} \ln \frac{n}{n}$

is divergent.

#### Explanation:

The function $\ln x$ is strictly increasing and as $\ln e = 1$ we have that $\ln n > 1$ for $n > 3$.

Therefore:

$\ln \frac{n}{n} > \frac{1}{n}$ for $n > 3$

and since ${\sum}_{n = 1}^{\infty} \frac{1}{n}$ is a divergent series then also

${\sum}_{n = 1}^{\infty} \ln \frac{n}{n}$

is divergent by direct comparison.