Harmonic series (mathematics)

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Harmonic series (mathematics)

See Harmonic series (music)
Harmonic series (music)

Definite pitch musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously....
for the (related) musical concept.

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the harmonic series is the divergent

Divergent series

In mathematics, a divergent series is an infinite series that is not Convergent series, meaning that the infinite sequence of the partial sums of the series does not have a limit of a sequence....
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

Harmonic mean

In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of Rate s is desired....
of the neighboring terms; the term harmonic mean likewise derives from music.

Divergence of the harmonic series
The harmonic series diverges

Divergent series

Infinity

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In mathematics

Mathematics

Divergent series

Harmonic mean

Divergence of the harmonic series

The harmonic series diverges

Divergent series

Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
, albeit rather slowly (the first 10⁴³ terms sum to less than 100 ). 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:

The sum of infinitely many terms clearly diverges to infinity and therefore the harmonic series also diverges. More precisely, if is the 2^k-th partial sum of the harmonic series, then

which clearly diverges, although slowly (at a logarithmic

Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
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

Cauchy condensation test

In mathematics, the Cauchy condensation test is a standard convergence test for infinite series. For a positive monotone decreasing sequence f, the sum...
is a generalization of this argument.

Another proof uses the integral test for convergence

Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. An early form of the test of convergence was developed in Indian mathematics by Madhava of Sangamagramma in the 14th century, and by his followers at the Kerala School....
, relating the harmonic series to the (divergent) integral of over the interval from 1 to infinity.

Even the sum of the reciprocals of just the prime number

Prime number

In mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC....
s diverges to infinity, although at an exponentially slower rate; known proofs of this fact

Proof that the sum of the reciprocals of the primes diverges

In the third century BC, Euclid proved the existence of infinitely many prime numbers. In the 18th century, Leonhard Euler proved a stronger statement: the sum of the multiplicative inverses of all prime numbers diverges, meaning that...
are much more difficult.

Alternate proof of divergence

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

Then, redistributing the fractions, leaves

Simplifying the second group yields

Substituting the second group for S leaves

This then comes to the conclusion that

or the conclusion

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

Convergence

In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium point state....
, and so must diverge

Divergent series

Convergence of the alternating harmonic series

The alternating harmonic series converges:

This equality is a consequence of the Mercator series

Mercator series

In mathematics, the Mercator series or Newton?Mercator series is the Taylor series for the natural logarithm. It is given byvalid for −1 < x ≤ 1....
, the Taylor series

Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
for the natural logarithm. Another equality, similar in form to Mercator's series, is:

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

Partial sums

The nth partial sum of the diverging harmonic series,

is called the nth harmonic number
Harmonic number

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural number:This also equals n times the inverse of the harmonic mean of these natural numbers....
.

The difference between the nth harmonic number and the natural logarithm

Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828....
of n converges to the Euler-Mascheroni constant

Euler-Mascheroni constant

The Euler?Mascheroni constant is a mathematical constant recurring in mathematical analysis and number theory, usually denoted by the lowercase Greek letter ....
.

The difference between distinct harmonic numbers is never an integer.

General harmonic series

The general harmonic series is of the form

All general harmonic series diverge.

P-series

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
Riemann zeta function

In mathematics, the Riemann zeta function, named after Germany mathematician Bernhard Riemann, is a prominent function of great significance in number theory because of its relation to the prime number theorem....
evaluated at p.

Random harmonic series

Byron Schmuland of the University of Alberta examined the properties of the random harmonic series

where the sn are independent
Statistical independence

In probability theory, to say that two event s are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs....
, identically distributed random variables taking the values +1 and −1 with equal probability 1/2. He shows that this sum converges with probability 1
Almost surely

In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory....
and that the convergent is a random variable
Random variable

In mathematics, random variables are used in the study of Randomness and probability. They were developed to assist in the analysis of Game of chance, stochastic events, and the results of experiment by capturing only the mathematical properties necessary to answer probability questions....
with some interesting properties. In particular, the probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
of this random variable evaluated at +2 or at −2 takes on the value , 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
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. In fact when terms containing any particular string of digits are removed the series converges.

See also

Harmonic series (music)
Harmonic series (music)

Definite pitch musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously....
Complex logarithm
Complex logarithm

In complex analysis, a complex logarithm function is an "inverse function" of the complex exponential function, just as the natural logarithm ln x is the inverse of the exponential function ex....
Harmonic number
Harmonic number

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural number:This also equals n times the inverse of the harmonic mean of these natural numbers....
Riemann zeta function
Riemann zeta function

In mathematics, the Riemann zeta function, named after Germany mathematician Bernhard Riemann, is a prominent function of great significance in number theory because of its relation to the prime number theorem....
Lagarias’s theorem
Riemann hypothesis

In mathematics, the Riemann hypothesis, due to , is a conjecture about the distribution of the Root of the Riemann zeta function stating that all non-trivial zeros of the Riemann zeta function have real part 1/2....

Many proofs of divergence of harmonic series : "", The AMATYC Review, 27 (2006), pp. 31-43.