In

^{43} terms is less than 100. This is because the partial sums of the series have

^{−42}. Schmuland's paper explains why this probability is so close to, but not exactly, . The exact value of this probability is given by the infinite cosine product integral divided by .

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, the harmonic series is the divergent infinite series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

: $\backslash sum\_^\backslash infty\backslash frac\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$
Its name derives from the concept of overtone
An overtone is any frequency
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is the indefinite continued sequence, progress of existence and event (philosophy) ...

s, or harmonics in music: the wavelength
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

s of the overtones of a vibrating string are , , , etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Sometimes it is appropriate for situations when the average rate (mathematics), rate is desired.
The harmonic mean can be express ...

of the neighboring terms; the phrase ''harmonic mean'' likewise derives from music.
History

The divergence of the harmonic series was first proven in the 14th century byNicole Oresme
Nicole Oresme (; c. 1320–1325 – July 11, 1382), also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme, was a significant philosopher
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the g ...

, but this achievement fell into obscurity. Proofs were given in the 17th century by and by Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss
Swiss may refer to:
* the adjectival form of Switzerland
,german: Schweizer(in),french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Svizra
, government ...

,
the latter proof published and popularized by his brother Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leibn ...

.
Historically, harmonic sequences have had a certain popularity with architects. This was so particularly in the Baroque
The Baroque (, ; ) is a of , , , , and other arts that flourished in Europe from the early 17th century until the 1740s. In the territories of the Spanish and Portuguese empires including the Iberian Peninsula it continued, together with new s ...

period, when architects used them to establish the proportions of floor plans
In architecture and building engineering, a floor plan is a drawing to Scale (ratio), scale, showing a view from above, of the relationships between rooms, spaces, traffic patterns, and other physical features at one level of a structure.
Dimensi ...

, of elevations
The elevation of a geographic location (geography), location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational equipotential surfa ...

, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.
Divergence

There are several well-known proofs of the divergence of the harmonic series. A few of them are given below.Comparison test

One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largestpower of two
Visualization of powers of two from 1 to 1024 (20 to 210)
A power of two is a number of the form where is an integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can b ...

:
:$\backslash begin\; 1\; \&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash frac\; \&\&\; +\; \backslash cdots\; \backslash \backslash $
\geq 1 & + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \cdots
\end
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 or equal to the sum of the second series. However, the sum of the second series is infinite:
:$\backslash begin\; \&\; 1\; +\; \backslash left(\backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\backslash right)\; +\; \backslash left(\backslash frac\; +\; \backslash cdots\; +\; \backslash frac\backslash right)\; +\; \backslash cdots\; \backslash \backslash $ = & 1 + \frac + \frac + \frac + \frac + \cdots = \infty.
\end
(Here, "$=\backslash infty$" is merely a notational convention to indicate that the partial sums of the series grow without bound.)
It follows (by the comparison test) that the sum of the harmonic series must be infinite as well. More precisely, the comparison above proves that
:$\backslash sum\_^\; \backslash frac\; \backslash geq\; 1\; +\; \backslash frac$
for every positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, in optics
* Plus sign, the sign "+" used to indicate a positive number
* Positive (electricity), a po ...

integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

.
This proof, proposed by 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
A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the g ...

in around 1350, is considered by many in the mathematical community to be a high point of . 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 possible to prove that the harmonic series diverges by comparing its sum with animproper integral
In mathematical analysis, an improper integral is the limit of a definite integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and units high, so the total area of the infinite number of rectangles is the sum of the harmonic series:
:$\backslash begin\; \backslash text\; \backslash \backslash \; \backslash text\; \backslash end\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$
Additionally, the total area under the curve from 1 to infinity is given by a divergent improper integral
In mathematical analysis, an improper integral is the limit of a definite integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

:
:$\backslash begin\; \backslash text\; \backslash \backslash \; \backslash text\backslash end\; =\; \backslash int\_1^\backslash infty\backslash frac\backslash ,dx\; =\; \backslash infty.$
Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, the first $k$ rectangles completely cover the region underneath the curve for $1\; \backslash leq\; x\; \backslash leq\; k\; +\; 1$ and so
$$\backslash sum\_^k\; \backslash frac\; >\; \backslash int\_1^\; \backslash frac\backslash ,dx\; =\; \backslash ln(k+1).$$
The generalization of this argument is known as the integral test
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

.
Rate of divergence

The harmonic series diverges very slowly. For example, the sum of the first 10logarithmic growth
A graph of logarithmic growth
In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. ''y'' = ''C'' log (''x''). Note that any logarithm base can be us ...

. In particular,
:$\backslash sum\_^k\backslash frac\; =\; \backslash ln\; k\; +\; \backslash gamma\; +\; \backslash varepsilon\_k\; \backslash leq\; (\backslash ln\; k)\; +\; 1$
where is the Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an Letter (alph ...

and which approaches 0 as goes to infinity. Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.
:$\backslash sum\_\backslash frac1p\; =\; \backslash frac12\; +\; \backslash frac13\; +\; \backslash frac15\; +\; \backslash frac17\; +\; \backslash frac1\; +\; \backslash frac1\; +\; \backslash frac1\; +\backslash cdots\; =\; \backslash infty.$
Partial sums

The finite partial sums of the diverging harmonic series, : $H\_n\; =\; \backslash sum\_^n\; \backslash frac,$ are calledharmonic number
In mathematics, the -th harmonic number is the sum of the Multiplicative inverse, reciprocals of the first natural numbers:
:H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
:1, \frac ...

s.
The difference between and converges to the Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an Letter (alph ...

. The difference between any two harmonic numbers is never an integer. No harmonic numbers are integers, except for .
Related series

Alternating harmonic series

The series : $\backslash sum\_^\backslash infty\; \backslash frac\; =\; 1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash cdots$ is known as the alternating harmonic series. This series converges by the alternating series test. In particular, the sum is equal to thenatural logarithm of 2The decimal value of the natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant
A mathematical constant is a key number
A number is a mathematical object used to counting, count, mea ...

:
:$1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash cdots\; =\; \backslash ln\; 2.$
The alternating harmonic series, while conditionally convergentIn mathematics, a series (mathematics), series or integral is said to be conditionally convergent if it converges, but it does not Absolute convergence, converge absolutely.
Definition
More precisely, a series of real numbers \sum_^\infty a_n is sai ...

, is not absolutely convergent
In mathematics, an Series (mathematics), infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a Real number, real or Complex number, ...

: if the terms in the series are systematically rearranged, in general the sum becomes different and, dependent on the rearrangement, possibly even infinite.
The alternating harmonic series formula is a special case of the Mercator series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, the Taylor series
In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...

for the natural logarithm.
A related series can be derived from the Taylor series for the arctangent
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

:
: $\backslash sum\_^\backslash infty\; \backslash frac\; =\; 1\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac\; +\; \backslash cdots\; =\; \backslash frac.$
This is known as the Leibniz series.
General harmonic series

The general harmonic series is of the form :$\backslash sum\_^\backslash frac\; ,$ where and are real numbers, and is not zero or a negative integer. By the limit comparison test with the harmonic series, all general harmonic series also diverge.-series

A generalization of the harmonic series is the -series (or hyperharmonic series), defined as :$\backslash sum\_^\backslash frac$ for any real number . When , the -series is the harmonic series, which diverges. Either theintegral test
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

or the Cauchy condensation test
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

shows that the -series converges for all (in which case it is called the over-harmonic series) and diverges for all . If then the sum of the -series is , i.e., the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter
The Greek alphabet has been used to write the Greek language
Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is ...

evaluated at .
The problem of finding the sum for is called the Basel problem
The Basel problem is a problem in mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebr ...

; Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

showed it is . The value of the sum for is called Apéry's constant
In mathematics, at the intersection of number theory and special functions, Apéry's constant is the summation, sum of the multiplicative inverse, reciprocals of the positive Cube (algebra), cubes. That is, it is defined as the number
:\begin\zet ...

, since Roger Apéry
Roger Apéry (; 14 November 1916, Rouen
Rouen (, ; or ) is a city on the River Seine in northern France. It is the capital of the region of Normandy
Normandy (; french: link=no, Normandie ; nrf, Normaundie; from Old French , plural of ...

proved that it is an irrational number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

.
ln-series

Related to the -series is the ln-series, defined as :$\backslash sum\_^\backslash frac$ for any positive real number . This can be shown by the integral test to diverge for but converge for all .-series

For anyconvex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optical device that focuses or disperses a light beam by means of ...

, real-valued function such that
:$\backslash limsup\_\backslash frac\; <\; \backslash frac,$
the series
:$\backslash sum\_^\backslash infty\; \backslash varphi\backslash left(\backslash frac\backslash right)$
is convergent.
Random harmonic series

The random harmonic series :$\backslash sum\_^\backslash frac,$ where the areindependent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independent ...

, identically distributed random variables taking the values +1 and −1 with equal is a well-known example in probability theory for a series of random variables that converges with probability 1. The fact of this convergence is an easy consequence of either the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. Byron Schmuland of the University of Alberta further examined the properties of the random harmonic series, and showed that the convergent series is a random variable
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...

with some interesting properties. In particular, the probability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...

of this random variable evaluated at +2 or at −2 takes on the value ..., differing from by less than 10Depleted harmonic series

The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value . In fact, when all the terms containing any particular string of digits (in anybase
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

) are removed, the series converges.
Applications

The harmonic series can becounterintuitive
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