There are several sub-types of harmonic series.
The the most basic harmonic series is the infinite sum This sum slowly approaches infinity.
The alternating harmonic series, , though, approaches .
The zeta-function is a harmonic series when the input is one.
How to solve
It can be shown that the harmonic series diverges by grouping the terms. We know that the first term, , added to the second term, is greater than . We also know that the third and and fourth terms, and , add up to something greater than . And we continue grouping the terms between powers of two. So we have
Alternating Harmonic Series
General Harmonic Series
is the general harmonic series, where each term is the reciprocal of a term in an arithmetic series.