The harmonic series is divergent

Posted June 28th, 2008 by Structure

The harmonic series is:

$1 + \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{n}+...$

This series is divergent.

Proof

$1 + \frac{1}{2}+ (\frac{1}{3}+ \frac{1}{4})+ (\frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8})+... >$

$1 + \frac{1}{2}+ (\frac{1}{4}+ \frac{1}{4})+ (\frac{1}{8}+ \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8})+... =$

$1 + \frac{1}{2}+ \frac{1}{2}+ \frac{1}{2}+ ... = +\infty$

Instant start