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 first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).
The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

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 the nth harmonic number and the natural logarithm of n converges to the Euler-Mascheroni constant.

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

Main article: Kempner 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] In fact when terms containing any particular string of digits are removed the series converges.

[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: Solution 72
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