9th  Differential_calculus">Top calculus topics: Differential calculus 
In calculus, Taylor's theorem gives a sequence of approximations of a differentiable function around a given point by polynomials (the Taylor polynomials of that function) whose coefficients depend only on the derivatives of the function at that point. The theorem also gives precise estimates on the size of the error in the approximation. The theorem is named after the mathematician Brook Taylor, who stated it in 1712, though the result was first discovered 41 years earlier in 1671 by James Gregory.
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Taylor's theorem asserts that any sufficiently smooth function can locally be approximated by polynomials. A simple example of application of Taylor's theorem is the approximation of the exponential function e^{x} near x = 0:
The approximation is called the nth order Taylor approximation to e^{x} because it approximates the value of the exponential function by a polynomial of degree n. This approximation only holds for x close to zero, and as x moves further away from zero, the approximation becomes worse. The quality of the approximation is controlled by the remainder term:
More generally, Taylor's theorem applies to any sufficiently differentiable function ƒ, giving an approximation, for x near a point a, of the form
The remainder term is just the difference of the function and its approximating polynomial
Although an explicit formula for the remainder term is seldom of any use, Taylor's theorem also provides several ways in which to estimate the value of the remainder. In other words, for x near enough to a, the remainder ought to be "small"; Taylor's theorem gives information on precisely how small it actually is.
The precise statement of the theorem is as follows: If n ≥ 0 is an integer and ƒ is a function which is n times continuously differentiable on the closed interval [a, x] and n + 1 times differentiable on the open interval (a, x), then
Here, n! denotes the factorial of n, and R_{n}(x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function. The remainder term R_{n}(x) depends on x and is small if x is close enough to a. Several expressions are available for it.
The Lagrange form^{[1]} of the remainder term states that there exists a number ξ between a and x such that
This exposes Taylor's theorem as a generalization of the mean value theorem. In fact, the mean value theorem is used to prove Taylor's theorem with the Lagrange remainder term.
The Cauchy form^{[2]} of the remainder term states that there exists a number ξ between a and x such that
More generally, if G(t) is a continuous function on [a,x] which is differentiable with nonvanishing derivative on (a,x), then there exists a number ξ between a and x such that
This exposes Taylor's theorem as a generalization of the Cauchy mean value theorem.
The above forms are restricted to the case of functions taking real values. However, the integral form^{[3]} of the remainder term applies as well when the function takes complex values. It is:
provided, as is often the case, ƒ^{(n)} is absolutely continuous on [a, x]. This shows the theorem to be a generalization of the fundamental theorem of calculus.
In general, a function does not need to be equal to its Taylor series, since it is possible that the Taylor series does not converge, or that it converges to a different function. However, for many functions ƒ(x), one can show that the remainder term R_{n} approaches zero as n approaches ∞. Those functions can be expressed as a Taylor series in a neighbourhood of the point a and are called analytic.
Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function ƒ has complex values or vector values. Furthermore, there is a version of Taylor's theorem for functions in several variables. For complex functions analytic in a region containing a circle C surrounding a and its interior, there is a contour integral expression for the remainder
valid inside of C.
Another common version of Taylor's theorem holds on an interval (a − r, a + r) where the variable x is assumed to take its values. This formulation of the theorem has the advantage that it is often possible to explicitly control the size of the remainder terms, and thus arrive at an approximation of a function valid in a whole interval with precise bounds on the quality of the approximation.
A precise version of Taylor's theorem in this form is as follows. Suppose ƒ is a function which is n times continuously differentiable on the closed interval [a − r, a + r] and n + 1 times differentiable on the open interval (a − r, a + r). If there exists a positive real constant M_{n} such that ƒ^{(n+1)}(x) ≤ M_{n} for all x ∈ (a − r, a + r), then
where the remainder function R_{n} satisfies the inequality (known as Cauchy's estimate):
for all x ∈ (a − r, a + r). This is called a uniform estimate of the error in the Taylor polynomial centered at a, because it holds uniformly for all x in the interval.
If ƒ is infinitely differentiable on [a − r, a + r], then positive constants M_{n} exist for each n = 1, 2, 3, … such that  ƒ^{(n+1)}(x) ≤ M_{n} for all x ∈ (a − r, a + r). If, in addition, it is possible to select these constants so that
then ƒ is an analytic function on (a − r, a + r). In particular, the remainder term in the Taylor approximation R_{n}(x) tends to zero uniformly as n→∞. In other words, an analytic function is the uniform limit of its Taylor polynomials on an interval.
Taylor's theorem can be generalized to several variables as follows. Let B be a ball in R^{N} centered at a point a, and ƒ be a realvalued function defined on the closure having n+1 continuous partial derivatives at every point. Taylor's theorem asserts that for any ,
where the summation extends over multiindices α (this formula uses the multiindex notation).
The remainder terms satisfy the inequality
for all α with α = n + 1. As was the case with one variable, the remainder terms can be described explicitly. See the proof for details.
We first prove Taylor's theorem with the integral remainder term.^{[4]}
The fundamental theorem of calculus states that
which can be rearranged to:
Now we can see that an application of Integration by parts yields:
The first equation is arrived at by letting and dv = dt; the second equation by noting that ; the third just factors out some common terms.
Another application yields:
By repeating this process, we may derive Taylor's theorem for higher values of n.
This can be formalized by applying the technique of induction. So, suppose that Taylor's theorem holds for a particular n, that is, suppose that
We can rewrite the integral using integration by parts. An antiderivative of (x − t)^{n} as a function of t is given by −(x−t)^{n+1} / (n + 1), so
Substituting this in (*) proves Taylor's theorem for n + 1, and hence for all nonnegative integers n.
The remainder term in the Lagrange form can be derived by the mean value theorem for integrals in the following way:
where ξ is some number from the interval [a, x]. The last integral can be evaluated immediately, which leads to
More generally, for any function G(t), the mean value theorem asserts the existence of ξ in the interval [a, x] satisfying
An alternative proof, which holds under milder technical assumptions on the function ƒ, can be supplied using the Cauchy mean value theorem. Let G be a realvalued function continuous on [a, x] and differentiable with nonvanishing derivative on (a, x). Let
By Cauchy's mean value theorem,
for some ξ ∈ (a, x). Note that the numerator F(x) − F(a) = R_{n} is the remainder of the Taylor polynomial for ƒ(x). On the other hand, computing F′(t),
Putting these two facts together and rearranging the terms of (1) yields
which was to be shown.
Note that the Lagrange form of the remainder comes from taking G(t) = (x − t)^{n+1}, the given Cauchy form of the remainder comes from taking G(t) = (t − a), and the integral form of the remainder comes from taking
Let x = (x_{1},...,x_{N}) lie in the ball B with center a. Parametrize the line segment between a and x by u(t) = a + t(x − a). We apply the onevariable version of Taylor's theorem to the function ƒ(u(t)):
By the chain rule for several variables,
where is the multinomial coefficient for the multiindex α. Since , we get
The remainder term is given by
The terms of this summation are explicit forms for the R_{α} in the statement of the theorem. These are easily seen to satisfy the required estimate.
