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

\sum_{k=1}^\infty \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots.\!

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.

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[edit] Divergence of the harmonic series

The harmonic series diverges, albeit rather slowly, to infinity (the first 1043 terms sum to less than 100 [1]). One way to prove this divergence is by noting that the harmonic series is term-by-term larger than or equal to another divergent series:

\begin{align} \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] +\cdots \\ & {} > 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] +\cdots \\ & {} = 1 + \ \frac{1}{2}\ \ \ + \quad \frac{1}{2} \ \quad + \ \qquad\quad\frac{1}{2}\qquad\ \quad \ + \quad \ \ \frac{1}{2} \ \quad +\cdots. \end{align}

The sum of infinitely many "1/2"s clearly diverges to infinity and therefore the harmonic series also diverges. More precisely, if s_{2^k} is the 2k-th partial sum of the harmonic series, then

s_{2^k} > 1 + {k \over 2} ,

which clearly diverges, although slowly (at a logarithmic rate).

This proof, due to Nicole Oresme, is a high point of medieval mathematics. It is still a standard proof taught in mathematics 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 fact are much more difficult.

[edit] Convergence of the alternating harmonic series

The alternating harmonic series converges:

\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2 = 0.6931471806\dots\,.

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:

\sum_{k=0}^\infty \frac{(-1)^k}{2k+1} = 1 - \frac{1}{3} + \frac{1}{5} -\frac{1}{7} +\cdots = \arctan (1 )=\frac{\pi}{4}.\!

This is a consequence of the Taylor series representation of the inverse tangent function (which has a radius of convergence of 1).

[edit] Partial sums

The nth partial sum of the diverging harmonic series,

H_n = \sum_{k = 1}^n \frac{1}{k},\!

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

\sigma(n)\le H_n + \ln(H_n)e^{H_n} \qquad \text{ for every }n \in \mathbb{N}, \!

where σ(n) stands for the sum of the positive divisors of n.[2]

[edit] General harmonic series

The general harmonic series is of the form

\sum_{n=0}^{\infty}\frac{1}{an+b}.\!

All general harmonic series diverge.

[edit] P-series

The p-series is (any of) the series

\sum_{n=1}^{\infty}\frac{1}{n^p},\!

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.

[edit] Random harmonic series

Byron Schmuland of the University of Alberta examined[3][4] the properties of the random harmonic series

\sum_{n=1}^{\infty}\frac{s_{n}}{n},\!

where the sn 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.

[edit] Depleted harmonic series

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.[5][6]

[edit] See also

[edit] Notes

  1. ^ On-Line Encyclopedia of Integer Sequences A082912
  2. ^ An Elementary Problem Equivalent to the Riemann Hypothesis, American Mathematical Monthly, volume 109 (2002), pages 534--543.
  3. ^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
  4. ^ Schmuland's preprint of Random Harmonic Series
  5. ^ Nick's Mathematical Puzzles: Answer 72
  6. ^ Nick's Mathematical Puzzles: Solution 72