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# Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent 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. [1]

Addison-Wesley is a publisher of textbooks and computer literature.

## Almost surely

In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one.

## Alternating series test

In mathematical analysis, the alternating series test is a method used to prove that an alternating series with terms that decrease in absolute value is a convergent series.

## Ant on a rubber rope

Ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical.

## Apéry's constant

In mathematics, at the crossing of number theory and special functions, Apéry's constant is defined as the number where ζ is the Riemann zeta function.

## Baroque

The Baroque is often thought of as a period of artistic style that used exaggerated motion and clear, easily interpreted detail to produce drama, tension, exuberance, and grandeur in sculpture, painting, architecture, literature, dance, theater, and music.

## Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1734 and read on 5 December 1735 in ''The Saint Petersburg Academy of Sciences'' (Петербургская Академия наук).

## Cauchy condensation test

In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series.

## Comparison test

Comparison test can mean.

## Complex logarithm

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

## Divergence of the sum of the reciprocals of the primes

The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.

## Divergent series

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

## Euler–Mascheroni constant

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (\gamma).

## Fundamental frequency

The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform.

## Harmonic mean

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and in particular one of the Pythagorean means.

## Harmonic number

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

## Harmonic progression (mathematics)

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression.

## Harmonic series (music)

A harmonic series is the sequence of all multiples of a base frequency.

## Improper integral

In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or \infty or -\infty or, in some cases, as both endpoints approach limits.

## Independence (probability theory)

In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of the other.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first, literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## 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.

## Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

## Jacob Bernoulli

Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family.

## Johann Bernoulli

Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family.

## Kolmogorov's inequality

In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound.

## Kolmogorov's three-series theorem

In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions.

## Leibniz formula for π

In mathematics, the Leibniz formula for pi, named after Gottfried Leibniz, states that Using summation notation.

## Leonhard Euler

Leonhard Euler (17071783) was a pioneering Swiss mathematician and physicist.

## List of sums of reciprocals

In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)&mdash;that is, it is generally the sum of unit fractions.

## Logarithmic growth

In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input.

## Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change.

## Mercator series

In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: In summation notation, The series converges to the natural logarithm (shifted by 1) whenever &minus;1 th derivative of ln x at x.

## Natural logarithm

The natural logarithm of a number is its logarithm to the base e, where ''e'' is an irrational and transcendental constant approximately equal to.

## Natural logarithm of 2

The decimal value of the natural logarithm of 2 is approximately as shown in the first line of the table below.

## Nicole Oresme

Nicole Oresme (c. 1320–1325 – July 11, 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a significant philosopher of the later Middle Ages.

## Overtone

An overtone is any frequency higher than the fundamental frequency of a sound.

A paradox is a statement that apparently contradicts itself and yet might be true (or wrong at the same time).

## Pietro Mengoli

Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647.

## Probability density function

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

## Proof without words

In mathematics, a proof without words is a proof of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.

## Proportion (architecture)

Proportion is a central principle of architectural theory.

## Random variable

In probability and statistics, a random variable, aleatory variable or stochastic variable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense).

## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1.

## Series (mathematics)

A series is, informally speaking, the sum of the terms of a sequence.

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number to be positive or negative.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## References

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