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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, ... .

Contents 1 Divergence of the harmonic series 2 Convergence of the alternating harmonic series 3 Harmonic numbers 4 General harmonic series 5 "p-series" 6 Random harmonic series 7 See also

[edit] Divergence of the harmonic series

The harmonic series diverges, albeit slowly, to infinity. One way to prove this is by noting that the harmonic series is term-by-term larger than or equal to another divergent series

<math>\sum_{k=1}^\infty \frac{1}{k} =

1 + \left[\frac{1}{2}\right] + \left[\frac{1}{3} + \frac{1}{4}\right] + \left[\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right] + \left[\frac{1}{9}+\cdots\right.</math>

<math> \quad\ \ge \sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil}\,\!</math>

<math> = 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] + \left[\frac{1}{16}+\cdots\right.\,\!</math>

<math> = 1 +\ \frac{1}{2}\ + \qquad\frac{1}{2} \ \quad+ \ \qquad\quad\frac{1}{2}\qquad\ \quad \ + \ \quad\ \cdots \,\!</math>

which clearly diverges. (Both sets of grouping can rigorously be imposed since all terms in each series have the same sign.) This proof, due to Nicole Oresme, is a high point of medieval mathematics.

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).

[edit] Convergence of the alternating harmonic series

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.

[edit] Harmonic numbers

If we define the nth harmonic number as

<math>H_n = \sum_{k = 1}^n \frac{1}{k}</math>

then H_n 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.

The difference between distinct harmonic numbers 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.)

[edit] General harmonic series

The general harmonic series is of the form

<math>\sum_{n=1}^{\infty}\frac{1}{an+b} </math>

All general harmonic series diverge.

[edit] "p-series"

The 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.

[edit] Random harmonic series

Byron Schmuland of the University of Alberta examined (American Mathematical Monthly 110, 407-416, May 2003; see [1]) the properties of the random harmonic series

<math>\sum_{n=1}^{\infty}\frac{s_{n}}{n} </math>

where the s_n are independent, identically distributed random variables taking the values +1 and −1 with equal probability 1/2. He shows that this sum converges with probability 1 and that the convergent is a random variable with some interesting properties. Of particular interest is that the probability density function of this random variable evaluated at +2 or at −2 takes on the value 0.1249999999999999999999999999999999999999997642..., differing from 1/8 by less than 10⁻⁴². Schmuland actually explains why it is so close to, but not exactly, 1/8.

[edit] See also

• Harmonic mean
• Harmonic number
• Riemann zeta functionca:Sèrie harmònica

da:Harmoniske række de:Harmonische Reihe es:Serie armónica (matemática) fr:Série harmonique he:הסדרה ההרמונית nl:Harmonische rij pl:Szereg harmoniczny pt:Série harmónica (matemática) sl:Harmonična vrsta sv:Harmoniska serien