Harmonic series (mathematics)
See harmonic series (music) for the (related) musical concept.
In mathematics, the harmonic series is the infinite series

It is so called because the wavelengths of the overtones of a vibrating string are proportional to 1, 1/2, 1/3, 1/4, ... .
It diverges, albeit slowly, to infinity. This can be proved by noting that the harmonic series is termbyterm larger than or equal to the series

which clearly diverges. Even the sum of the reciprocals of the
prime numbers diverges to infinity (although that is much harder to prove;
see here).
The
alternating harmonic series converges however:

This is a consequence of the
Taylor series of the
natural logarithm.
If we define the nth harmonic number as

then
H_{n} grows about as fast as the
natural logarithm of . The reason is that the sum is approximated by the
integral

whose value is ln(
n).
More precisely, we have the limit:

where γ is the
EulerMascheroni constant.
Lagarias proved in 2001 that the Riemann hypothesis is equivalent to the statement

where σ(
n) stands for the sum of positive divisors of
n.
The generalised harmonic series, or pseries, is (any of) the series
for
p a positive real number. The series is convergent if
p>1 and divergent otherwise. When
p=1, the series is the harmonic series. If
p > 1 then the sum of the series is ζ(
p), i.e., the
Riemann zeta function evaluated at
p.
This can be used in the testing of convergence of series.