Taylor's Theorem Taylor Series Taylor County Florida

In calculus, Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, allows the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. The precise statement is as follows: If n≥0 is an integer and f 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 we have

Here, n! denotes the factorial of n, and R is a remainder term which depends on x and is small if x is close enough to a. Several expressions for R are available.

The Lagrange form 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.

For some functions f(x), one can show that the remainder term R approaches zero as n approaches ∞; those functions can be expressed as a Taylor series in a neighborhood of the point a and are called analyticIn mathematics, an analytic function is one that is locally given by a convergent power series. Complex analysis teaches us that if a function f of one complex variable is differentiable in some open disk D centered at a point c in the complex field, then.

Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function f has complexThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher values or vectorThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for values. Furthermore, there is a version of Taylor's theorem for functions in several variables.

Proof

We first prove Taylor's theorem with the integral remainder term.

The fundamental theorem of calculusThe fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. This means that if a continuous function is first integrated and then differentiated, the origin states that

This proves the theorem for n = 0.

Integration by partsCalculus In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. The rule arises from the product rule of differentiation. yields the case n = 1

By repeating this process, we may derive Taylor's theorem for higher values of n.

This can be formalized by applying the technique of inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence. A somewhat more general form of argument used in mathe. So, suppose that Taylor's theorem holds for a particular n, that is, suppose that

We can again rewrite the integral using integration by partsCalculus In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. The rule arises from the product rule of differentiation.. 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 Lagrangian form can be derived by the mean value theorem in the following way:

The last integral can be solved immediately, which leads to

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