# Math Help - Show that this series diverges

1. ## Show that this series diverges

Hi, how do I show that this series is divergent?

$\sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)$

Thanks.

2. Originally Posted by coldfire
Hi, how do I show that this series is divergent?

$\sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)$

Thanks.
I suggest using a ratio test and then a series of log rules to simplify the expression.

3. Originally Posted by coldfire
Hi, how do I show that this series is divergent?

$\sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)$

Thanks.
Note that

$\ln \left( \frac {n+1}{n}\right) = \ln (n+1) - \ln n$

so the series is telescopic.

4. Hello, coldfire

Show that this series is divergent:

. . $\sum_{n=1}^{\infty}\ln\left(\frac{n+1}{n} \right)$

We have: . $\sum^{\infty}_{n=1}\ln\left(\frac{n+1}{n}\right)$

. . . . . . $=\;\;\lim_{n\to\infty}\bigg[\ln\left(\frac{2}{1}\right) + \ln\left(\frac{3}{2}\right) + \ln\left(\frac{4}{3}\right) + \ln\left(\frac{5}{4}\right) + \hdots + \ln\left(\frac{n+1}{n}\right)\bigg]$

. . . . . . $=\;\;\lim_{n\to\infty}\bigg[\ln\left(\frac{\rlap{/}2}{1}\cdot\frac{\rlap{/}3}{\rlap{/}2}\cdot\frac{\rlap{/}4}{\rlap{/}3}\cdot\frac{\rlap{/}5}{\rlap{/}4}\:\cdots\:\frac{n+1}{\rlap{/}n}\right)\bigg]$

. . . . . . $=\;\;\lim_{n\to\infty}\ln\,(n+1) \;\;=\;\;\ln(\infty) \;\;=\;\;\infty$