How do you show that the harmonic series diverges?

Question

Axel Alvarez · Accepted Answer

undefined The harmonic series diverges, which means its sum approaches infinity as you add more terms. This can be shown using the integral test or by comparing it to other known divergent series, such as the series of reciprocals of natural numbers. The integral test involves comparing the series to the integral of 1/x from 1 to infinity, which diverges logarithmically. Alternatively, you can demonstrate divergence by grouping terms in the series into sets, such as powers of 2, and showing that each set sums to a value greater than 1.

Evelyn Agard · Answer

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.