Harmonic series (mathematics)

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

In mathematics, the harmonic series is the divergent 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.

[edit] Divergence of the harmonic series

The harmonic series diverges to infinity, albeit rather slowly (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 terms clearly diverges to infinity and therefore the harmonic series also diverges. More precisely, if $s_{2^k}$ is the 2^k-th partial sum of the harmonic series, then

$s_{2^k} \ge 1 + {k \over 2},$

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. Cauchy's condensation test is a generalization of this argument.

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] Alternate proof of divergence

Suppose that the Harmonic series converges to a sum, S:

$\sum_{k=1}^\infty \frac{1}{k} = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots=S$

Then, redistributing the fractions, leaves

$S=\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\right)+\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots\right)$

Simplifying the second group yields

$S=\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\right)+\frac{1}{2}\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots\right)$

Substituting the second group for S leaves

$S=\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\right)+\frac{1}{2}S$

This then comes to the conclusion that

$\frac{1}{2}S=\left(1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\right)$

or the conclusion

$\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots= 1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots$

This certainly cannot be true, as one is larger than one half, one third is larger than one fourth, etc, so the sum cannot converge, and so must diverge.^[2]

[edit] Another proof of divergence

Start with a basic geometric series:

$\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + ...$

Integrate both sides to obtain

$-\ln(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + ...$

Taking the limit of both sides as $x \rightarrow 1$ we have:

$-\lim_{x\to 1} \ln(1-x) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... = \sum_{n=1}^{\infty} \frac{1}{n}$ .

Since $-\lim_{x\to 1} \ln(1-x) = -(-\infty) = \infty$ , we see that the sum $\sum_{n=1}^{\infty} \frac{1}{n} = \infty$

In other words, the sum diverges.

[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 alternating harmonic series converges:

$\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2 = 0.693\,147\,180\,\dots.$

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.

[edit] General harmonic series

The general harmonic series is of the form

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

where the constants a and b are each any finite real number.

All general harmonic series diverge. ^[3]

[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 integral test shows that the series always converges if p > 1 (in which case it is called the over-harmonic series) and diverges otherwise. When p = 1, the series is the harmonic series, which diverges. 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^[4]^[5] the properties of the random harmonic series

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

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 0.124 999 999 999 999 999 999 999 999 999 999 999 999 999 764 …, 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.^[6] In fact when terms containing any particular string of digits are removed the series converges.

[edit] See also

Harmonic series (music)
Complex logarithm
Harmonic number
Riemann zeta function
Lagarias’s theorem
Many proofs of divergence of harmonic series : "The Harmonic Series Diverges Again and Again", The AMATYC Review, 27 (2006), pp. 31-43.

[edit] Notes

^ Sequence A082912 in the On-Line Encyclopedia of Integer Sequences
^ Nahin, Paul J. Dr. Euler's Fabulous Formula: cures many mathematical ills. Princeton: Princeton University Press, 2006.
^ Art of Problem Solving: "General Harmonic Series"
^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
^ Schmuland's preprint of Random Harmonic Series
^ Nick's Mathematical Puzzles: Solution 72

[0] Sequence A082912 in the On-Line Encyclopedia of Integer Sequences

[1] Nahin, Paul J. Dr. Euler's Fabulous Formula: cures many mathematical ills. Princeton: Princeton University Press, 2006.

[2] Art of Problem Solving: "General Harmonic Series"

[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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[6]

Harmonic series (mathematics)

From Wikipedia, the free encyclopedia

Contents

[edit] Divergence of the harmonic series

[edit] Alternate proof of divergence

[edit] Another proof of divergence

[edit] Convergence of the alternating harmonic series

[edit] Partial sums

[edit] General harmonic series

[edit] P-series

[edit] Random harmonic series

[edit] Depleted harmonic series

[edit] See also

[edit] Notes

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