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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 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