# How do you show that the series 1/2+2/3+3/4+...+n/(n+1)+... diverges?

Mar 1, 2017

Is divergent.

#### Explanation:

$\frac{1}{n} < \frac{n}{n + 1}$ and

${\sum}_{k = 1}^{\infty} \frac{1}{k} < {\sum}_{k = 1}^{\infty} \frac{k}{k + 1}$

but ${\sum}_{k = 1}^{\infty} \frac{1}{k}$ is the so called harmonic series which is divergent. So ${\sum}_{k = 1}^{\infty} \frac{k}{k + 1}$ is also divergent.