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