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 term-by-term 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 n-th 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
Euler-Mascheroni 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 p-series, 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.
See also
