Why does the Harmonic Series diverge?

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Why does the Harmonic Series diverge?

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Answer 1 · 2014-09-20T11:20:24

The harmonic series diverges.
$\infty \sum n = 1 \frac{1}{n} = \infty$ $sum_{n=1}^{infty}1/n=infty$

Let us show this by the comparison test.
$\infty \sum n = 1 \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \dots$ $sum_{n=1}^{infty}1/n=1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+cdots$
by grouping terms,
$= 1 + \frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) + (\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}) + \dots$ $=1+1/2+(1/3+1/4)+(1/5+1/6+1/7+1/8)+cdots$
by replacing the terms in each group by the smallest term in the group,
$> 1 + \frac{1}{2} + (\frac{1}{4} + \frac{1}{4}) + (\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}) + \dots$ $>1+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+cdots$
$= 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \dots$ $=1+1/2+1/2+1/2+cdots$
since there are infinitly many $\frac{1}{2}$ $1/2$ 's,
$= \infty$ $=infty$

Since the above shows that the harmonic series is larger that the divergent series, we may conclude that the harmonic series is also divergent by the comparison test.

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Why does the Harmonic Series diverge?

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