The harmonic series is divergent

The harmonic series is:

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

This series is divergent.

Proof

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