Harmonic series: Information from Answers.com

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

In mathematics, the harmonic series is the divergent infinite series:

$\sum_{n=1}^\infty\,\frac{1}{n} \;\;=\;\; 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.\!$

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.

1 History
2 Divergence
3 Partial sums
4 Related series
5 See also
6 References
7 External links

History

The fact that the harmonic series diverges was first proven in the 14^th century by Nicole Oresme, but this was mislaid. Later proofs were found by Pietro Mengoli, Johann Bernoulli and Jakob Bernoulli in the 17^th century.

Divergence

The harmonic series diverges to infinity. There are several well-known proofs of this fact.

Comparison test

One way to prove divergence is to compare the harmonic series with another divergent series:

$\begin{align} & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{3} \,+\, \frac{1}{4} \;\;+\;\; \frac{1}{5} \,+\, \frac{1}{6} \,+\, \frac{1}{7} \,+\, \frac{1}{8} \;\;+\;\; \frac{1}{9} \,+\, \cdots \\[12pt] >\;\;\; & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{4} \,+\, \frac{1}{4} \;\;+\;\; \frac{1}{8} \,+\, \frac{1}{8} \,+\, \frac{1}{8} \,+\, \frac{1}{8} \;\;+\;\; \frac{1}{16} \,+\, \cdots \;\;=\;\; \infty. \end{align}$

Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than the sum of the second series. However, the sum of the second series is infinite:

$\begin{align} & 1 + \left(\frac{1}{2}\right) + \left(\frac{1}{4}+\frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \left(\frac{1}{16}+\cdots+\frac{1}{16}\right) + \cdots \\[12pt] =\;\; & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \cdots \;\;=\;\; \infty. \end{align}$

It follows (by the comparison test) that sum of the harmonic series must be infinite as well. More precisely, the comparison above proves that

$\sum_{n=1}^{2^k} \,\frac{1}{n} \;>\; 1 + \frac{k}{2}$

for every positive integer k. 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.

Integral test

It is also possible to prove that the harmonic series diverges by comparing the sum with an improper integral. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and 1 / n units high, so the total area of the rectangles is the sum of the harmonic series:

$\begin{array}{c} \text{area of}\\ \text{rectangles} \end{array} = 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.$

However, the total area under the curve y = 1 / x from 1 to infinity is given by an improper integral:

$\begin{array}{c} \text{area under}\\ \text{curve} \end{array} = \int_1^\infty\frac{1}{x}\,dx \;=\; \infty.$

Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, this proves that

$\sum_{n=1}^k \, \frac{1}{n} \;>\; \int_1^{k+1} \frac{1}{x}\,dx \;=\; \ln(k+1).$

The generalization of this argument is known as the integral test.

Rate of divergence

The harmonic series diverges very slowly. For example, the sum of the first 10⁴³ terms is less than 100 ^[1]. This is because the partial sums of the series have logarithmic growth. In particular,

$\sum_{n=1}^k\,\frac{1}{n} \;=\; \ln k + \gamma + \varepsilon_k$

where $γ$ is the Euler–Mascheroni constant and $ε k$ approaches 0 as $k$ goes to infinity. This result is due to Leonhard Euler.

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.

Related series

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 series

$\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} \;=\; 1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots$

is known as the alternating harmonic series. This series converges by the alternating series test. In particular, the sum is equal to the natural logarithm of 2:

$1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots \;=\; \ln 2.$

This formula is a special case of the Mercator series, the Taylor series for the natural logarithm.

A related series can be derived from the Taylor series for the arctangent:

$\sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} \;\;=\;\; 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \cdots \;\;=\;\; \frac{\pi}{4}.$

This is known as the Leibniz formula for pi.

General harmonic series

The general harmonic series is of the form

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

where $a \ne 0$ and $b$ are real numbers.

By the comparison test, all general harmonic series diverge. ^[2]

P-series

A generalisation of the harmonic series is the p-series, defined as:

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

for any positive real number p. When p = 1, the p-series is the harmonic series, which diverges. Either the integral test or the Cauchy condensation test shows that the p-series converges for all p > 1 (in which case it is called the over-harmonic series) and diverges for all p ≤ 1. If p > 1 then the sum of the p-series is ζ(p), i.e., the Riemann zeta function evaluated at p.

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 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 0.124999999999999999999999999999999999999999764…, 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.

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.

References

^ Sequence A082912 in the On-Line Encyclopedia of Integer Sequences
^ 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

External links

"The Harmonic Series Diverges Again and Again", The AMATYC Review, 27 (2006), pp. 31-43. Many proofs of divergence of harmonic series.

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Harmonic series

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