Harmonic Series
Key Questions

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.

Since the harmonic series is known to diverge, we can use it to compare with another series. When you use the comparison test or the limit comparison test, you might be able to use the harmonic series to compare in order to establish the divergence of the series in question.

The harmonic series is
#sum_{n=1}^infty 1/n=1+1/2+1/3+1/4+cdots#
(Note: This is a divergent series.)
Questions
Tests of Convergence / Divergence

Geometric Series

Nth Term Test for Divergence of an Infinite Series

Direct Comparison Test for Convergence of an Infinite Series

Ratio Test for Convergence of an Infinite Series

Integral Test for Convergence of an Infinite Series

Limit Comparison Test for Convergence of an Infinite Series

Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

Infinite Sequences

Root Test for for Convergence of an Infinite Series

Infinite Series

Strategies to Test an Infinite Series for Convergence

Harmonic Series

Indeterminate Forms and de L'hospital's Rule

Partial Sums of Infinite Series