Question

How do you show that the harmonic series diverges?

Answer 1

The harmonic series diverges.
`sum_{n=1}^{infty}1/n=infty`

Let us show this by the comparison test.
`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+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+1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)+cdots`
`=1+1/2+1/2+1/2+cdots`
since there are infinitly many `1/2`'s,
`=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.