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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the Taylor series of a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
is an
infinite sum In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of terms that are expressed in terms of the function's
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
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
Brook Taylor Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Life and work Brook Taylor ...
, who introduced them in 1715. If 0 is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after
Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish Scottish usually refers to something of, from, or related to Scotland, including: *Scottish Gaelic, a Celtic Goidelic language of the Indo-Europ ...
, who made extensive use of this special case of Taylor series in the 18th century. The
partial sum 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 ...
formed by the first terms of a Taylor series is a
polynomial 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 ...

polynomial
of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases.
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is
convergent Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to: *Convergent boundary, a type of plate tectonic boundary * Convergent (continued fraction) * Convergent evolution * Convergent s ...
, its sum is the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of the
infinite sequence 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 ...
of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point if it is equal to the sum of its Taylor series in some
open interval 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 ...
(or
open disk In geometry, a disk (also Spelling of disc, spelled disc). is the region in a plane (geometry), plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. Form ...
in the
complex plane 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 ...
) containing . This implies that the function is analytic at every point of the interval (or disk).


Definition

The Taylor series of a
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematic ...
that is
infinitely differentiable is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...
at a
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
is the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
:f(a)+\frac (x-a)+ \frac (x-a)^2+\frac(x-a)^3+ \cdots, where denotes the
factorial 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 g ...
of . In the more compact
sigma notation 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 ...
, this can be written as : \sum_ ^ \frac (x-a)^, where denotes the th
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
of evaluated at the point . (The derivative of order zero of is defined to be itself and and are both defined to be 1.) When , the series is also called a
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1 ...
.


Examples

The Taylor series for any
polynomial 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 ...

polynomial
is the polynomial itself. The Maclaurin series for is the
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
:1 + x + x^2 + x^3 + \cdots, so the Taylor series for at is :1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots. By integrating the above Maclaurin series, we find the Maclaurin series for , where denotes the
natural logarithm The natural logarithm of a number is its logarithm 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 ( ...
: :-x - \tfracx^2 - \tfracx^3 - \tfracx^4 - \cdots. The corresponding Taylor series for at is :(x-1) - \tfrac(x-1)^2 + \tfrac(x-1)^3 - \tfrac(x-1)^4 + \cdots, and more generally, the corresponding Taylor series for at an arbitrary nonzero point is: :\ln a + \frac (x - a) - \frac\frac + \cdots. The Maclaurin series for the
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

exponential function
is :\begin \sum_^\infty \frac &= \frac + \frac + \frac + \frac + \frac + \frac+ \cdots \\ &= 1 + x + \frac + \frac + \frac + \frac + \cdots. \end The above expansion holds because the derivative of with respect to is also , and equals 1. This leaves the terms in the numerator and in the denominator for each term in the infinite sum.


History

The Greek philosopher
Zeno Zeno or Zenon ( grc, Ζήνων) may refer to: People * Zeno (name), including a list of people and characters with the name Philosophers * Zeno of Elea (), philosopher, follower of Parmenides, known for his paradoxes * Zeno of Citium (333 – 2 ...

Zeno
considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was
Zeno's paradox Zeno's paradoxes are a set of philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, existence, knowledge Knowledge is a familiarity, awareness, or understanding of someo ...
. Later,
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

Aristotle
proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
, as it had been prior to Aristotle by the Presocratic Atomist
Democritus Democritus (; el, Δημόκριτος, ''Dēmókritos'', meaning "chosen of the people"; – ) was an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient w ...

Democritus
. It was through Archimedes's
method of exhaustion The method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
that an infinite number of progressive subdivisions could be performed to achieve a finite result.
Liu Hui Liu Hui () was a Chinese mathematician and writer who lived in the state of Cao Wei Wei (220–266), also known as Cao Wei or Former Wei, was one of the three major states that competed for supremacy over China in the Three Kingdoms perio ...
independently employed a similar method a few centuries later. In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by
Madhava of Sangamagrama Iriññāttappiḷḷi Mādhavan Nampūtiri known as Mādhava of Sangamagrāma () was an Indian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes t ...
. Though no record of his work survives, writings of later
Indian mathematicians The chronology of Indian mathematicians spans from the Indus Valley Civilization oxen for pulling a cart and the presence of the chicken The chicken (''Gallus gallus domesticus''), a subspecies of the red junglefowl, is a type of d ...
suggest that he found a number of special cases of the Taylor series, including those for the
trigonometric function 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 ...

trigonometric function
s of
sine 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). ...

sine
,
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
,
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
, and
arctangent In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

arctangent
. The
Kerala School of Astronomy and Mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
further expanded his works with various
series expansion 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 ...
s and rational approximations until the 16th century. In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by
Brook Taylor Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Life and work Brook Taylor ...
, after whom the series are now named. The Maclaurin series was named after
Colin Maclaurin Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish Scottish usually refers to something of, from, or related to Scotland, including: *Scottish Gaelic, a Celtic Goidelic language of the Indo-Europ ...
, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.


Analytic functions

If is given by a convergent power series in an open disk (or interval in the real line) centred at in the complex plane, it is said to be analytic in this disk. Thus for in this disk, is given by a convergent power series :f(x) = \sum_^\infty a_n(x-b)^n. Differentiating by the above formula times, then setting gives: :\frac = a_n and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centred at if and only if its Taylor series converges to the value of the function at each point of the disk. If is equal to the sum of its Taylor series for all in the complex plane, it is called
entire *In philately, see Cover (philately), Cover *In mathematics, see Entire function *In animal fancy and animal husbandry, entire (animal), entire indicates that an animal has not been desexed, that is, spayed or neutered *In botany, an edge (such as o ...
. The polynomials,
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

exponential function
, and the
trigonometric function 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 ...

trigonometric function
s sine and cosine, are examples of entire functions. Examples of functions that are not entire include the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
, the
logarithm 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 ...

logarithm
, the
trigonometric function 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 ...

trigonometric function
tangent, and its inverse,
arctan 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 ...

arctan
. For these functions the Taylor series do not
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines See also

...
if is far from . That is, the Taylor series diverges at if the distance between and is larger than the
radius of convergence 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 gener ...
. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions include: # The partial sums (the
Taylor polynomial 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). ...
s) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included. #Differentiation and integration of power series can be performed term by term and is hence particularly easy. #An
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is uniquely extended to a
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
on an open disk in the
complex plane 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 ...

complex plane
. This makes the machinery of
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
available. #The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). #Algebraic operations can be done readily on the power series representation; for instance,
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

Euler's formula
follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as
harmonic analysis Harmonic analysis is a branch of 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 (mathe ...
. #Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.


Approximation error and convergence

Pictured on the right is an accurate approximation of around the point . The pink curve is a polynomial of degree seven: :\sin\left( x \right) \approx x - \frac + \frac - \frac.\! The error in this approximation is no more than . In particular, for , the error is less than 0.000003. In contrast, also shown is a picture of the natural logarithm function and some of its Taylor polynomials around . These approximations converge to the function only in the region ; outside of this region the higher-degree Taylor polynomials are ''worse'' approximations for the function. The ''error'' incurred in approximating a function by its th-degree Taylor polynomial is called the ''remainder'' or '' residual'' and is denoted by the function . Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be
convergent Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to: *Convergent boundary, a type of plate tectonic boundary * Convergent (continued fraction) * Convergent evolution * Convergent s ...
at all. And in fact the set of functions with a convergent Taylor series is a
meager set In the fields of and , a meagre set (also called a meager set or a set of first category) is a that, considered as a of a (usually larger) , is in a precise sense small or . A topological space is called meagre if it is a meager subset of its ...
in the
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are Complete space, complete with ...
of
smooth functions is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over ...
. And even if the Taylor series of a function does converge, its limit need not in general be equal to the value of the function . For example, the function : f(x) = \begin e^ & \text x \neq 0 \\
mu
mu
0 & \text x = 0 \end is
infinitely differentiable is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...
at , and has all derivatives zero there. Consequently, the Taylor series of about is identically zero. However, is not the zero function, so does not equal its Taylor series around the origin. Thus, is an example of a
non-analytic smooth function
non-analytic smooth function
. In
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

real analysis
, this example shows that there are
infinitely differentiable function In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mat ...
s whose Taylor series are ''not'' equal to even if they converge. By contrast, the
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s studied in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
always possess a convergent Taylor series, and even the Taylor series of
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function (mathematics), function that is holomorphic function, holomorphic on all of ''D'' ''except'' for a set of isolated p ...
s, which might have singularities, never converge to a value different from the function itself. The complex function , however, does not approach 0 when approaches 0 along the imaginary axis, so it is not
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
in the complex plane and its Taylor series is undefined at 0. More generally, every sequence of real or complex numbers can appear as
coefficient 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 ...
s in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of
Borel's lemma In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations. Statement Suppose ''U'' is an open set in the Euclidean space R''n'', and suppose that ' ...
. As a result, the
radius of convergence 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 gener ...
of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere. A function cannot be written as a Taylor series centred at a
singularity Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiabl ...
; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable ; see
Laurent series (Holomorphic functions are analytic, analytic). In mathematics, the Laurent series of a complex function ''f''(''z'') is a representation of that function as a power series which includes terms of negative degree. It may be used to express compl ...

Laurent series
. For example, can be written as a Laurent series.


Generalization

There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded
continuous function 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 gen ...
on , using the calculus of
finite differences A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for th ...
. Specifically, one has the following theorem, due to
Einar Hille
Einar Hille
, that for any , :\lim_\sum_^\infty \frac\frac = f(a+t). Here is the th finite difference operator with step size . The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the
Newton series A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
. When the function is analytic at , the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series. In general, for any infinite sequence , the following power series identity holds: :\sum_^\infty\frac\Delta^na_i = e^\sum_^\infty\fraca_. So in particular, :f(a+t) = \lim_ e^\sum_^\infty f(a+jh) \frac. The series on the right is the
expectation value In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum p ...
of , where is a Poisson-distributed
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 ...
that takes the value with probability . Hence, :f(a+t) = \lim_ \int_^\infty f(a+x)dP_(x). The
law of large numbers
law of large numbers
implies that the identity holds.


List of Maclaurin series of some common functions

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments .


Exponential function

The
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

exponential function
e^x (with base ) has Maclaurin series :e^ = \sum^_ \frac = 1 + x + \frac + \frac + \cdots . It converges for all .


Natural logarithm

The
natural logarithm The natural logarithm of a number is its logarithm 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 ( ...
(with base ) has Maclaurin series :\begin \ln(1-x) &= - \sum^_ \fracn = -x - \frac2 - \frac3 - \cdots , \\ \ln(1+x) &= \sum^\infty_ (-1)^\fracn = x - \frac2 + \frac3 - \cdots . \end They converge for , x, < 1. (In addition, the series for converges for , and the series for converges for .)


Geometric series

The
geometric series In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
and its derivatives have Maclaurin series :\begin \frac &= \sum^\infty_ x^n \\ \frac &= \sum^\infty_ nx^\\ \frac &= \sum^\infty_ \frac x^. \end All are convergent for , x, < 1. These are special cases of the
binomial series The binomial series is the Taylor 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, ...
given in the next section.


Binomial series

The
binomial series The binomial series is the Taylor 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, ...
is the power series :(1+x)^\alpha = \sum_^\infty \binom x^n whose coefficients are the generalized
binomial coefficient 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 g ...
s : \binom = \prod_^n \frack = \frac. (If , this product is an
empty product 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 ...
and has value 1.) It converges for , x, < 1 for any real or complex number . When , this is essentially the infinite geometric series mentioned in the previous section. The special cases and give the
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
function and its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...

inverse
: :\begin (1+x)^\frac12 &= 1 + \tfracx - \tfracx^2 + \tfracx^3 - \tfracx^4 + \tfracx^5 - \ldots &&=\sum^_ \frac x^n, \\ (1+x)^ &= 1 -\tfracx + \tfracx^2 - \tfracx^3 + \tfracx^4 - \tfracx^5 + \ldots &&=\sum^_ \frac x^n. \end When only the linear term is retained, this simplifies to the
binomial approximation The binomial approximation is useful for approximately calculating powers Powers (stylized as POWERS) is a musical duo composed of Mike Del Rio and Crista Ru. Their music has been described as alternative pop, electropop, and Progressive p ...
.


Trigonometric functions

The usual
trigonometric function 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 ...

trigonometric function
s and their inverses have the following Maclaurin series: :\begin \sin x &= \sum^_ \frac x^ &&= x - \frac + \frac - \cdots && \text x\\
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\cos x &= \sum^_ \frac x^ &&= 1 - \frac + \frac - \cdots && \text x\\
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\tan x &= \sum^_ \frac x^ &&= x + \frac + \frac + \cdots && \text, x, < \frac\\
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\sec x &= \sum^_ \frac x^ &&=1+\frac+\frac+\cdots && \text, x, < \frac\\
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\arcsin x &= \sum^_ \frac x^ &&=x+\frac+\frac+\cdots && \text, x, \le 1\\
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\arccos x &=\frac-\arcsin x\\&=\frac- \sum^_ \frac x^&&=\frac-x-\frac-\frac-\cdots&& \text, x, \le 1\\
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\arctan x &= \sum^_ \frac x^ &&=x-\frac + \frac-\cdots && \text, x, \le 1,\ x\neq\pm i \end All angles are expressed in
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s. The numbers appearing in the expansions of are the
Bernoulli numbers 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). ...
. The in the expansion of are
Euler number 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 ...
s.


Hyperbolic functions

The
hyperbolic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

hyperbolic function
s have Maclaurin series closely related to the series for the corresponding trigonometric functions: :\begin \sinh x &= \sum^_ \frac &&= x + \frac + \frac + \cdots && \text x\\
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\cosh x &= \sum^_ \frac &&= 1 + \frac + \frac + \cdots && \text x\\
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\tanh x &= \sum^_ \frac x^ &&= x-\frac+\frac-\frac+\cdots && \text, x, < \frac\\
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\operatorname x &= \sum^_ \frac x^ &&=x - \frac + \frac - \cdots && \text, x, \le 1\\
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\operatorname x &= \sum^_ \frac &&=x + \frac + \frac +\cdots && \text, x, \le 1,\ x\neq\pm 1 \end The numbers appearing in the series for are the
Bernoulli numbers 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). ...
.


Calculation of Taylor series

Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying
integration by parts In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

integration by parts
. Particularly convenient is the use of
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data. It is a type of applica ...

computer algebra system
s to calculate Taylor series.


First example

In order to compute the 7th degree Maclaurin polynomial for the function :f(x)=\ln(\cos x),\quad x\in\left(-\frac,\frac\right) , one may first rewrite the function as :f(x)=\ln\bigl(1+(\cos x-1)\bigr)\!. The Taylor series for the natural logarithm is (using the
big O notation Big O notation is a mathematical notation that describes the limiting behavior of a function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be ...
) :\ln(1+x) = x - \frac2 + \frac3 + \left(x^4\right)\! and for the cosine function :\cos x - 1 = -\frac2 + \frac - \frac + \left(x^8\right)\!. The latter series expansion has a zero
constant term 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 ...
, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big notation: :\beginf(x)&=\ln\bigl(1+(\cos x-1)\bigr)\\ &=(\cos x-1) - \tfrac12(\cos x-1)^2 + \tfrac13(\cos x-1)^3+ \left((\cos x-1)^4\right)\\ &=\left(-\frac2 + \frac - \frac +\left(x^8\right)\right)-\frac12\left(-\frac2+\frac+\left(x^6\right)\right)^2+\frac13\left(-\frac2+O\left(x^4\right)\right)^3 + \left(x^8\right)\\ & =-\frac2 + \frac-\frac - \frac8 + \frac - \frac +O\left(x^8\right)\\ & =- \frac2 - \frac - \frac+O\left(x^8\right). \end\! Since the cosine is an
even function The cosine function and all of its Taylor polynomials are even functions. This image shows \cos(x) and its Taylor approximation of degree 4. In mathematics, even functions and odd functions are function (mathematics), functions which satisfy par ...

even function
, the coefficients for all the odd powers have to be zero.


Second example

Suppose we want the Taylor series at 0 of the function : g(x)=\frac.\! We have for the exponential function : e^x = \sum^\infty_ \frac =1 + x + \frac + \frac + \frac+\cdots\! and, as in the first example, : \cos x = 1 - \frac + \frac - \cdots\! Assume the power series is : \frac = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots\! Then multiplication with the denominator and substitution of the series of the cosine yields : \begin e^x &= \left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots\right)\cos x\\ &=\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - \frac + \frac - \cdots\right)\\&=c_0 - \fracx^2 + \fracx^4 + c_1x - \fracx^3 + \fracx^5 + c_2x^2 - \fracx^4 + \fracx^6 + c_3x^3 - \fracx^5 + \fracx^7 + c_4x^4 +\cdots \end\! Collecting the terms up to fourth order yields : e^x =c_0 + c_1x + \left(c_2 - \frac\right)x^2 + \left(c_3 - \frac\right)x^3+\left(c_4-\frac+\frac\right)x^4 + \cdots\! The values of c_i can be found by comparison of coefficients with the top expression for e^x, yielding: : \frac=1 + x + x^2 + \frac + \frac + \cdots.\!


Third example

Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand as a Taylor series in , we use the known Taylor series of function : : e^x = \sum^\infty_ \frac =1 + x + \frac + \frac + \frac+\cdots. Thus, : \begin(1+x)e^x &= e^x + xe^x = \sum^\infty_ \frac + \sum^\infty_ \frac = 1 + \sum^\infty_ \frac + \sum^\infty_ \frac \\ &= 1 + \sum^\infty_ \frac + \sum^\infty_ \frac =1 + \sum^\infty_\left(\frac + \frac\right)x^n \\ &= 1 + \sum^\infty_\fracx^n\\ &= \sum^\infty_\fracx^n.\end


Taylor series as definitions

Classically,
algebraic functionIn 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 ha ...
s are defined by an algebraic equation, and
transcendental function 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 ...
s (including those discussed above) are defined by some property that holds for them, such as a
differential equation In mathematics, a differential equation is an equation 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 ( ...

differential equation
. For example, the
exponential function The exponential function is a mathematical function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of ...

exponential function
is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an
analytic function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
by its Taylor series. Taylor series are used to define functions and " operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the
matrix exponential 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 ...
or
matrix logarithm In mathematics, a logarithm of a matrix is another matrix (mathematics), matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse funct ...
. In other areas, such as formal analysis, it is more convenient to work directly with the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
themselves. Thus one may define a solution of a differential equation ''as'' a power series which, one hopes to prove, is the Taylor series of the desired solution.


Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with :\begin T(x_1,\ldots,x_d) &= \sum_^\infty \cdots \sum_^\infty \frac\,\left(\frac\right)(a_1,\ldots,a_d) \\ &= f(a_1, \ldots,a_d) + \sum_^d \frac (x_j - a_j) + \frac \sum_^d \sum_^d \frac (x_j - a_j)(x_k - a_k) \\ & \qquad \qquad + \frac \sum_^d\sum_^d\sum_^d \frac (x_j - a_j)(x_k - a_k)(x_l - a_l) + \cdots \end For example, for a function f(x,y) that depends on two variables, and , the Taylor series to second order about the point is :f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) + \frac\Big( (x-a)^2 f_(a,b) + 2(x-a)(y-b) f_(a,b) +(y-b)^2 f_(a,b) \Big) where the subscripts denote the respective
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

partial derivative
s. A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as :T(\mathbf) = f(\mathbf) + (\mathbf - \mathbf)^\mathsf D f(\mathbf) + \frac (\mathbf - \mathbf)^\mathsf \left \ (\mathbf - \mathbf) + \cdots, where is the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

gradient
of evaluated at and is the
Hessian matrix In mathematic Mathematics (from Greek: ) includes the study of such topics as quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in te ...
. Applying the
multi-index notation Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index Index may refer to: Arts, ...
the Taylor series for several variables becomes :T(\mathbf) = \sum_\frac \left(f\right)(\mathbf), which is to be understood as a still more abbreviated
multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus Calculus, originally called infinitesimal calcu ...
version of the first equation of this paragraph, with a full analogy to the single variable case.


Example

In order to compute a second-order Taylor series expansion around point of the function :f(x,y)=e^x\ln(1+y), one first computes all the necessary partial derivatives: :\begin f_x &= e^x\ln(1+y) \\
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f_y &= \frac \\
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f_ &= e^x\ln(1+y) \\
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f_ &= - \frac \\
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f_ &=f_ = \frac . \end Evaluating these derivatives at the origin gives the Taylor coefficients :\begin f_x(0,0) &= 0 \\ f_y(0,0) &=1 \\ f_(0,0) &=0 \\ f_(0,0) &=-1 \\ f_(0,0) &=f_(0,0)=1. \end Substituting these values in to the general formula :\begin T(x,y) = &f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) \\ &+\frac\left( (x-a)^2f_(a,b) + 2(x-a)(y-b)f_(a,b) +(y-b)^2 f_(a,b) \right)+ \cdots \end produces :\begin T(x,y) &= 0 + 0(x-0) + 1(y-0) + \frac\Big( 0(x-0)^2 + 2(x-0)(y-0) + (-1)(y-0)^2 \Big) + \cdots \\ &= y + xy - \frac + \cdots \end Since is analytic in , we have :e^x\ln(1+y)= y + xy - \frac + \cdots, \qquad , y, < 1.


Comparison with Fourier series

The trigonometric
Fourier series 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 gen ...
enables one to express a
periodic function A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ...

periodic function
(or a function defined on a closed interval ) as an infinite sum of
trigonometric function 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 ...

trigonometric function
s (
sine 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). ...

sine
s and
cosine In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...

cosine
s). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of
powers Powers (stylized as POWERS) is a musical duo composed of Mike Del Rio and Crista Ru. Their music has been described as alternative pop, electropop, and Progressive pop, progressive pop. ''Time'' has called their music "groovy and futuristic". ...

powers
. Nevertheless, the two series differ from each other in several relevant issues: * The finite truncations of the Taylor series of about the point are all exactly equal to at . In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact. * The computation of Taylor series requires the knowledge of the function on an arbitrary small
neighbourhood A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain interval. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global". * The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any
integrable function 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). ...
. In particular, the function could be nowhere differentiable. (For example, could be a
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Weierstrass function
.) * The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges
pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
to the function, and uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is
square-integrable 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 ...
then the series converges in
quadratic mean 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 ...
, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C1 then the convergence is uniform). * Finally, in practice one wants to approximate the function with a finite number of terms, say with a Taylor polynomial or a partial sum of the trigonometric series, respectively. In the case of the Taylor series the error is very small in a neighbourhood of the point where it is computed, while it may be very large at a distant point. In the case of the Fourier series the error is distributed along the domain of the function.


See also

* Asymptotic expansion * Generating function * Madhava series * Newton polynomial, Newton's divided difference interpolation * Padé approximant * Puiseux series * Shift operator


Notes


References

* * *


External links

* * {{Authority control Real analysis Complex analysis Series expansions