Taylor's theorem

In calculus, Taylor's theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives 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.

Related Topics:
Calculus - Mathematician - Brook Taylor - 1712 - Differentiable - Function - Polynomial

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The most basic example is the approximation of the exponential function extrm{e}^x near x = 0. Namely,

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: extrm{e}^x pprox 1 + x + rac{x^2}{2!} + rac{x^3}{3!} + cdots + rac{x^N}{N!}.

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The precise statement of the theorem is as follows: If n ≥ 0 is an integer and f is a function which is n times continuously differentiable on the closed interval and n + 1 times differentiable on the open interval (a, x), then we have

Related Topics:
Integer - Closed interval

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: f(x) = f(a) + rac{f'(a)}{1!}(x - a) + rac{f^{(2)}(a)}{2!}(x - a)^2 + cdots + rac{f^{(n)}(a)}{n!}(x - a)^n + R_n

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Here, n! denotes the factorial of n, and Rn is a remainder term which depends on x and is small

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if x is close enough to a. Several expressions for Rn are available.

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The Lagrange form of the remainder term states that there exists a number ξ between a and x such that

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:

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R_n = rac{f^{(n+1)}(xi)}{(n+1)!} (x-a)^{n+1}.

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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.

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The Cauchy form of the remainder term is

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:

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R_n(x) = int_a^x rac{f^{(n+1)} (t)}{n!} (x - t)^n , dt.

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This shows the theorem to be a generalization of the fundamental theorem of calculus.

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For some functions f(x), one can show that the remainder term Rn 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.

Related Topics:
Taylor series - Neighbourhood - Analytic

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Taylor's theorem (with the integral formulation of the remainder term) is also valid if the function f has complex values or vector values. Furthermore, there is a version of Taylor's theorem for functions in several variables.

Related Topics:
Complex - Vector

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For complex functions analytic in a region containing a circle C surrounding a and its interior, we have a contour integral expression for the remainder

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: R_n(x) = rac{1}{2 pi i}int_C rac{f(z)}{(z-a)^{n+1}(z-x)}dz

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valid inside of C.

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