Hi, how do I show that this series is divergent?
$\displaystyle \sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)$
Thanks.
Hello, coldfire
Show that this series is divergent:
. . $\displaystyle \sum_{n=1}^{\infty}\ln\left(\frac{n+1}{n} \right)$
We have: .$\displaystyle \sum^{\infty}_{n=1}\ln\left(\frac{n+1}{n}\right)$
. . . . . . $\displaystyle =\;\;\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] $
. . . . . . $\displaystyle =\;\;\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] $
. . . . . . $\displaystyle =\;\;\lim_{n\to\infty}\ln\,(n+1) \;\;=\;\;\ln(\infty) \;\;=\;\;\infty $