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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor 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. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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 gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the infinite sequence 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 Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
at a point if it is equal to the sum of its Taylor series in some
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
(or open disk in the complex plane) containing . This implies that the function is analytic at every point of the interval (or disk).


Definition

The Taylor series of a real or complex-valued function that is infinitely differentiable at a real or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
:f(a)+\frac (x-a)+ \frac (x-a)^2+\frac(x-a)^3+ \cdots, where denotes the factorial of . In the more compact sigma notation, this can be written as : \sum_ ^ \frac (x-a)^, where denotes the th
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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.


Examples

The Taylor series of any
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
is the polynomial itself. The Maclaurin series of is the geometric series :1 + x + x^2 + x^3 + \cdots. So, by substituting for , the Taylor series of at is :1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots. By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the natural logarithm: :-x - \tfracx^2 - \tfracx^3 - \tfracx^4 - \cdots. The corresponding Taylor series of at is :(x-1) - \tfrac(x-1)^2 + \tfrac(x-1)^3 - \tfrac(x-1)^4 + \cdots, and more generally, the corresponding Taylor series of at an arbitrary nonzero point is: :\ln a + \frac (x - a) - \frac\frac + \cdots. The Maclaurin series of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
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 of each term in the infinite sum.


History

The ancient Greek philosopher Zeno of Elea 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 problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plur ...
. Later,
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
, 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 pre-Socratic philosopher from Abdera, primarily remembered today for his formulation of an atomic theory of the universe. No ...
. It was through Archimedes's
method of exhaustion The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in are ...
that an infinite number of progressive subdivisions could be performed to achieve a finite result.
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
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. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the
trigonometric function 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 a ...
s of sine, cosine,
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
, and
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
. Madhava founded the Kerala school of astronomy and mathematics, and during the following two centuries its scholars developed further series expansions and rational approximations. 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, a professor in Edinburgh, who published the special case of the Taylor result in the mid-18th century.


Analytic functions

If is given by a convergent power series in an open disk centred at in the complex plane (or an interval in the real line), it is said to be
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
in this region. Thus for in this region, 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 Entire may refer to: * Entire function, a function that is holomorphic on the whole complex plane * Entire (animal), an indication that an animal is not neutered * Entire (botany) This glossary of botanical terms is a list of definitions of ...
. The polynomials,
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, and the
trigonometric function 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 a ...
s sine and cosine, are examples of entire functions. Examples of functions that are not entire include the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
, the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
, the
trigonometric function 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 a ...
tangent, and its inverse,
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
. For these functions the Taylor series do not converge if is far from . That is, the Taylor series diverges at if the distance between and is larger than the radius of convergence. 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 polynomials) 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, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis 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 In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. Note that this paper is written in terms of the ''Shifted'' Chebyshev polynomials of the ...
). #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 mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis. #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 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 . For a full cycle centered at the origin () the error is less than 0.08215. 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 at all. And in fact the set of functions with a convergent Taylor series is a
meager set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
in the Fréchet space of smooth functions. 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 0 & \text x = 0 \end is infinitely differentiable 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 In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is ...
. In real analysis, this example shows that there are infinitely differentiable functions whose Taylor series are ''not'' equal to even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, 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 g ...
in the complex plane and its Taylor series is undefined at 0. More generally, every sequence of real or complex numbers can appear as coefficients 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 th ...
. As a result, the radius of convergence 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; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable ; see 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, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
on , using the calculus of finite differences. Specifically, one has the following theorem, due to
Einar Hille Carl Einar Hille (28 June 1894 – 12 February 1980) was an American mathematics professor and scholar. Hille authored or coauthored twelve mathematical books and a number of mathematical papers. Early life and education Hille was born in New ...
, 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. 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 of , where is a
Poisson-distributed In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known ...
random variable that takes the value with probability . Hence, :f(a+t) = \lim_ \int_^\infty f(a+x)dP_(x). The 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 denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
e^x (with base ) has Maclaurin series :e^ = \sum^_ \frac = 1 + x + \frac + \frac + \cdots . It converges for all . The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function: :\exp exp(x)-1= \sum_^ \fracx^


Natural logarithm

The natural logarithm (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 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 given in the next section.


Binomial series

The binomial series is the power series (1+x)^\alpha = \sum_^\infty \binom x^n whose coefficients are the generalized
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s \binom = \prod_^n \frack = \frac. (If , this product is an empty product 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, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
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 a ...
: \begin (1+x)^\frac &= 1 + \tfracx - \tfracx^2 + \tfracx^3 - \tfracx^4 + \tfracx^5 - \cdots &&=\sum^_ \frac x^n, \\ (1+x)^ &= 1 -\tfracx + \tfracx^2 - \tfracx^3 + \tfracx^4 - \tfracx^5 + \cdots &&=\sum^_ \frac x^n. \end When only the linear term is retained, this simplifies to the binomial approximation.


Trigonometric functions

The usual
trigonometric function 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 a ...
s and their inverses have the following Maclaurin series: :\begin \sin x &= \sum^_ \frac x^ &&= x - \frac + \frac - \cdots && \text x\\ pt\cos x &= \sum^_ \frac x^ &&= 1 - \frac + \frac - \cdots && \text x\\ pt\tan x &= \sum^_ \frac x^ &&= x + \frac + \frac + \cdots && \text, x, < \frac\\ pt\sec x &= \sum^_ \frac x^ &&=1+\frac+\frac+\cdots && \text, x, < \frac\\ pt\arcsin x &= \sum^_ \frac x^ &&=x+\frac+\frac+\cdots && \text, x, \le 1\\ pt\arccos x &=\frac-\arcsin x\\&=\frac- \sum^_ \frac x^&&=\frac-x-\frac-\frac-\cdots&& \text, x, \le 1\\ pt\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 rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s. The numbers appearing in the expansions of are the Bernoulli numbers. The in the expansion of are Euler numbers.


Hyperbolic functions

The
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
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\\ pt\cosh x &= \sum^_ \frac &&= 1 + \frac + \frac + \cdots && \text x\\ pt\tanh x &= \sum^_ \frac x^ &&= x-\frac+\frac-\frac+\cdots && \text, x, < \frac\\ pt\operatorname x &= \sum^_ \frac x^ &&=x - \frac + \frac - \cdots && \text, x, \le 1\\ pt\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.


Polylogarithmic functions

The polylogarithms have these defining identities: :\text_(x) = \sum_^ \frac x^ :\text_(x) = \sum_^ \frac x^ The Legendre chi functions are defined as follows: :\chi_(x) = \sum_^ \frac x^ :\chi_(x) = \sum_^ \frac x^ And the formulas presented below are called ''inverse tangent integrals'': :\text_(x) = \sum_^ \frac x^ :\text_(x) = \sum_^ \frac x^ In statistical thermodynamics these formulas are of great importance.


Elliptic functions

The complete elliptic integrals of first kind K and of second kind E can be defined as follows: :\fracK(x) = \sum_^ \fracx^ :\fracE(x) = \sum_^ \fracx^ The
Jacobi theta functions In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field the ...
describe the world of the elliptic modular functions and they have these Taylor series: :\vartheta_(x) = 1 + 2\sum_^ x^ :\vartheta_(x) = 1 + 2\sum_^ (-1)^ x^ The regular partition number sequence P(n) has this generating function: :\vartheta_(x)^\vartheta_(x)^\biggl frac\biggr = \sum_^ P(n)x^n = \prod_^ \frac The strict partition number sequence Q(n) has that generating function: :\vartheta_(x)^\vartheta_(x)^\biggl frac\biggr = \sum_^ Q(n)x^n = \prod_^ \frac


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, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
. Particularly convenient is the use of
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
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 when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...
) :\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, 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 In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
, 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 functions are defined by an algebraic equation, and
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed ...
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 that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
. For example, the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
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, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
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 or matrix logarithm. 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 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
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, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
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, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of evaluated at and is the Hessian matrix. Applying the multi-index notation 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 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) \\ ptf_y &= \frac \\ ptf_ &= e^x\ln(1+y) \\ ptf_ &= - \frac \\ ptf_ &=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 A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
enables one to express a
periodic function A periodic function is a 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 throughout science to des ...
(or a function defined on a closed interval ) as an infinite sum of
trigonometric function 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 a ...
s ( sines and cosines). 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. 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, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
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 particular, the function could be nowhere differentiable. (For example, could be a
Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstr ...
.) * 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
pointwise In 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 ...
to the function, and uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is square-integrable then the series converges in
quadratic mean In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
, 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 * Laurent series *
Madhava series In mathematics, a Madhava series or Leibniz series is any one of the series in a collection of infinite series expressions all of which are believed to have been discovered by an Indian Mathematician and Astronomer Madhava of Sangamagrama (c.&nbs ...
* Newton's divided difference interpolation * Padé approximant * Puiseux series * Shift operator


Notes


References

* * *


External links

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