How do you show that the harmonic series diverges?

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.