
In
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, Taylor's theorem gives an approximation of a
-times
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
around a given point by a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
of degree
, called the
-th-order Taylor polynomial. For a
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
, the Taylor polynomial is the truncation at the order ''
'' of the
Taylor series
In 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 derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the function. The first-order Taylor polynomial is the
linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.
Taylor's theorem is named after the mathematician
Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in
1671
Events
January–March
* January 1 – The Criminal Ordinance of 1670, the first attempt at a uniform code of criminal procedure in France, goes into effect after having been passed on August 26, 1670.
* January 5 – The ...
by
James Gregory.
Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
. It gives simple arithmetic formulas to accurately compute values of many
transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction ...
s such as the
exponential function and
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 all ...
s.
It is the starting point of the study of
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 ...
s, and is fundamental in various areas of mathematics, as well as in
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
and
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
. Taylor's theorem also generalizes to
multivariate and
vector valued functions. It provided the mathematical basis for some landmark early computing machines:
Charles Babbage
Charles Babbage (; 26 December 1791 – 18 October 1871) was an English polymath. A mathematician, philosopher, inventor and mechanical engineer, Babbage originated the concept of a digital programmable computer.
Babbage is considered ...
's
difference engine
A difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was created by Charles Babbage. The name ''difference engine'' is derived from the method of finite differen ...
calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.
Motivation
If a real-valued
function is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
at the point
, then it has a
linear approximation near this point. This means that there exists a function ''h''
1(''x'') such that
Here
is the linear approximation of
for ''x'' near the point ''a'', whose graph
is the
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the graph
at . The error in the approximation is:
As ''x'' tends to ''a,'' this error goes to zero much faster than
, making
a useful approximation.
For a better approximation to
, we can fit a
quadratic polynomial
In mathematics, a quadratic function of a single variable is a function of the form
:f(x)=ax^2+bx+c,\quad a \ne 0,
where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather tha ...
instead of a linear function:
Instead of just matching one derivative of
at
, this polynomial has the same first and second derivatives, as is evident upon differentiation.
Taylor's theorem ensures that the ''quadratic approximation'' is, in a sufficiently small neighborhood of
, more accurate than the linear approximation. Specifically,
Here the error in the approximation is
which, given the limiting behavior of
, goes to zero faster than
as ''x'' tends to ''a''.

Similarly, we might get still better approximations to ''f'' if we use
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s of higher degree, since then we can match even more derivatives with ''f'' at the selected base point.
In general, the error in approximating a function by a polynomial of degree ''k'' will go to zero much faster than
as ''x'' tends to ''a''. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be
analytic at ''x = a'': it is not (locally) determined by its derivatives at this point.
Taylor's theorem is of asymptotic nature: it only tells us that the error
in an
approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...
by a
-th order Taylor polynomial ''P
k'' tends to zero faster than any nonzero
-th degree
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
as
. It does not tell us how large the error is in any concrete
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions on ''f''. These enhanced versions of Taylor's theorem typically lead to
uniform estimates for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function ''f'' is
analytic. In that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)
There are several ways we might use the remainder term:
# Estimate the error for a polynomial ''P
k''(''x'') of degree ''k'' estimating
on a given interval (''a'' – ''r'', ''a'' + ''r''). (Given the interval and degree, we find the error.)
# Find the smallest degree ''k'' for which the polynomial ''P
k''(''x'') approximates
to within a given error tolerance on a given interval (''a'' − ''r'', ''a'' + ''r'') . (Given the interval and error tolerance, we find the degree.)
# Find the largest interval (''a'' − ''r'', ''a'' + ''r'') on which ''P
k''(''x'') approximates
to within a given error tolerance. (Given the degree and error tolerance, we find the interval.)
Taylor's theorem in one real variable
Statement of the theorem
The precise statement of the most basic version of Taylor's theorem is as follows:
The polynomial appearing in Taylor's theorem is the
-th order Taylor polynomial
of the function ''f'' at the point ''a''. The Taylor polynomial is the unique "asymptotic best fit" polynomial in the sense that if there exists a function and a
-th order polynomial ''p'' such that
then ''p'' = ''P
k''. Taylor's theorem describes the asymptotic behavior of the remainder term
which is the
approximation error
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation derived for it. This inherent error in approximation can be quantified and express ...
when approximating ''f'' with its Taylor polynomial. Using the
little-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 German mathematicians Pau ...
, the statement in Taylor's theorem reads as
Explicit formulas for the remainder
Under stronger regularity assumptions on ''f'' there are several precise formulas for the remainder term ''R
k'' of the Taylor polynomial, the most common ones being the following.
These refinements of Taylor's theorem are usually proved using the
mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
, whence the name. Additionally, notice that this is precisely the
mean value theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin ...
when
. Also other similar expressions can be found. For example, if ''G''(''t'') is continuous on the closed interval and differentiable with a non-vanishing derivative on the open interval between
and
, then
for some number
between
and
. This version covers the Lagrange and Cauchy forms of the remainder as special cases, and is proved below using
Cauchy's mean value theorem. The Lagrange form is obtained by taking
and the Cauchy form is obtained by taking
.
The statement for the integral form of the remainder is more advanced than the previous ones, and requires understanding of
Lebesgue integration theory for the full generality. However, it holds also in the sense of
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...
provided the (''k'' + 1)th derivative of ''f'' is continuous on the closed interval
'a'',''x''
Due to the
absolute continuity
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between ...
of ''f'' on the
closed interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
between
and
, its derivative ''f'' exists as an ''L''-function, and the result can be
proven by a formal calculation using the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
and
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 antiderivati ...
.
Estimates for the remainder
It is often useful in practice to be able to estimate the remainder term appearing in the Taylor approximation, rather than having an exact formula for it. Suppose that ''f'' is -times continuously differentiable in an interval ''I'' containing ''a''. Suppose that there are real constants ''q'' and ''Q'' such that
throughout ''I''. Then the remainder term satisfies the inequality
if , and a similar estimate if . This is a simple consequence of the Lagrange form of the remainder. In particular, if
on an interval with some
, then
for all The second inequality is called a
uniform estimate, because it holds uniformly for all ''x'' on the interval
Example

Suppose that we wish to find the approximate value of the function
on the interval