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See Harmonic series (music) for the (related) musical concept.

In mathematics, the harmonic series is the infinite series

Its name derives from the concept of overtones, or harmonics, in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the term harmonic mean likewise derives from music.

Contents 1 Divergence of the harmonic series 2 Convergence of the alternating harmonic series 3 Partial sums 4 General harmonic series 5 P-series 6 Random harmonic series 7 Depleted harmonic series 8 See also 9 Notes

The harmonic series diverges, albeit rather slowly, to infinity (the first 10⁴³ terms sum to less than 100). One way to prove the divergence is by noting that the harmonic series is term-by-term larger than or equal to another divergent series

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. It is still the standard proof taught in classes today.

Another proof uses the integral test for convergence, relating the harmonic series to the (divergent) integral of 1/x over the interval from 1 to infinity.

Even the sum of the reciprocals of just the prime numbers diverges to infinity, although at an exponentially slower rate; known proofs of this are much more difficult.

The alternating harmonic series converges:

This equality is a consequence of the Mercator series, the Taylor series for the natural logarithm. Another equality, similar in form to Mercator's series, is:

The nth partial sum of the diverging harmonic series,

is called the nth harmonic number.

The difference between distinct harmonic numbers is never an integer.

Jeffrey Lagarias proved in 2001 that the Riemann hypothesis is logically equivalent to the statement

where σ(n) stands for the sum of the positive divisors of n.^[1]

The general harmonic series is of the form

All general harmonic series diverge.

The p-series is (any of) the series

for any positive real number p. The series is always convergent if p > 1 (in which case it is called the over-harmonic series) 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.

Byron Schmuland of the University of Alberta examined^[2][3] the properties of the random harmonic series

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. In particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value 0.124 999 999 999 999 999 999 999 999 999 999 999 999 999 7642 ..., differing from 1/8 by less than 10⁻⁴². Schmuland's paper explains why this probability is so close to, but not exactly, 1/8.

The depleted harmonic series where all of the terms with a 9 in the denominator are removed can be shown to converge and its value is less than 80.^[4][5]

• Complex logarithm
• Harmonic number
• Riemann zeta function

1. ^ An Elementary Problem Equivalent to the Riemann Hypothesis, American Mathematical Monthly, volume 109 (2002), pages 534--543.
2. ^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
3. ^ Schmuland's preprint of Random Harmonic Series
4. ^ http://www.qbyte.org/puzzles/p072a.html
5. ^ http://www.qbyte.org/puzzles/p072s.html