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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 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 ...
: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \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) ...

overtone
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 ...

wavelength
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 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 ...
, but this achievement fell into obscurity. Proofs were given in the 17th century by
Pietro Mengoli
Pietro Mengoli
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 ...

Johann Bernoulli
, 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 ...

Baroque
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 ...

floor plans
, 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 ...

elevations
, 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-largest
power 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 ...
: :\begin 1 & + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \frac && + \cdots \\
pt
pt
\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: :\begin & 1 + \left(\frac\right) + \left(\frac + \frac\right) + \left(\frac + \frac + \frac + \frac\right) + \left(\frac + \cdots + \frac\right) + \cdots \\
pt
pt
= & 1 + \frac + \frac + \frac + \frac + \cdots = \infty. \end (Here, "=\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 :\sum_^ \frac \geq 1 + \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
medieval mathematics
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 possible to prove that the harmonic series diverges by comparing its sum with an
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 ...

improper integral
. 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: :\begin \text \\ \text \end = 1 + \frac + \frac + \frac + \frac + \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 ...

improper integral
: :\begin \text \\ \text\end = \int_1^\infty\frac\,dx = \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 \leq x \leq k + 1 and so \sum_^k \frac > \int_1^ \frac\,dx = \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 1043 terms is less than 100. This is because the partial sums of the series have
logarithmic 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, :\sum_^k\frac = \ln k + \gamma + \varepsilon_k \leq (\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 ...

Leonhard Euler
proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e. :\sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 +\cdots = \infty.


Partial sums

The finite partial sums of the diverging harmonic series, : H_n = \sum_^n \frac, are called
harmonic 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 : \sum_^\infty \frac = 1 - \frac + \frac - \frac + \frac - \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 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 - \frac + \frac - \frac + \frac - \cdots = \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 ...

Mercator series
, 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 ...

arctangent
: : \sum_^\infty \frac = 1 - \frac + \frac - \frac + \cdots = \frac. This is known as the Leibniz series.


General harmonic series

The general harmonic series is of the form :\sum_^\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 :\sum_^\frac for any real number . When , the -series is the harmonic series, which diverges. Either 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). ...
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 ...

Riemann zeta function
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 ...

Leonhard Euler
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 ...

Apéry's constant
, 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 :\sum_^\frac for any positive real number . This can be shown by the integral test to diverge for but converge for all .


-series

For any
convex 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 ...

convex
, real-valued function such that :\limsup_\frac < \frac, the series :\sum_^\infty \varphi\left(\frac\right) is convergent.


Random harmonic series

The random harmonic series :\sum_^\frac, where the are
independent 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 10−42. 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 .


Depleted 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 any
base 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 be
counterintuitive {{Short pages monitor