Harmonic series (mathematics)
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See harmonic series (music) for the (related) musical concept.
In mathematics, the harmonic series is the infinite series
- <math>\sum_{k=1}^\infty \frac{1}{k} =
1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots <math>
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
- <math>\sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil} \! =
1 + \left[\frac{1}{2}\right] + \left[\frac{1}{4} + \frac{1}{4}\right] + \left[\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right] + \frac{1}{16}\cdots <math>
- <math> = \quad\ 1 +\ \frac{1}{2}\ +\ \quad\frac{1}{2} \ \quad+ \ \qquad\quad\frac{1}{2}\qquad\ \quad \ + \ \quad\ \cdots <math>
which clearly diverges. Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see proof that the sum of the reciprocals of the primes diverges). The alternating harmonic series converges however:
- <math>\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k} = \ln 2.<math>
This is a consequence of the Taylor series of the natural logarithm.
If we define the n-th harmonic number as
- <math>H_n = \sum_{k = 1}^n \frac{1}{k}<math>
then Hn grows about as fast as the natural logarithm of n. The reason is that the sum is approximated by the integral
- <math>\int_1^n {1 \over x}\, dx<math>
whose value is ln(n).
More precisely, we have the limit:
- <math> \lim_{n \to \infty} H_n - \ln(n) = \gamma<math>
where γ is the Euler-Mascheroni constant.
It has been proven that:
- The only Hn that is an integer is H1.
- The difference Hm - Hn where m>n is never an integer.
Jeffrey Lagarias proved in 2001 that the Riemann hypothesis is equivalent to the statement
- <math>\sigma(n)\le H_n + \ln(H_n)e^{H_n} \qquad \mbox{ for every }n\in\mathbb{N}<math>
where σ(n) stands for the sum of positive divisors of n. (See An Elementary Problem Equivalent to the Riemann Hypothesis, American Mathematical Monthly, volume 109 (2002), pages 534--543.)
The generalised harmonic series, or p-series, is (any of) the series
- <math>\sum_{n=1}^{\infty}\frac{1}{n^p} <math>
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
es:Serie armónica (matemáticas) nl:Harmonische rij pl:Szereg harmoniczny pt:Série harmónica (matemática)