Taylor's theorem is named after the mathematician Brook Taylor (1685--1731), who stated (without a proof) a version of it in 1715. A general formulation of Taylor's theorem was given by the Swiss mathematician Johann Bernoulli (1667--1748) who published this statement in 1694. Brook Taylor did not entirely appreciate its larger importance of Taylor's theorem and he certainly did not bother with a formal proof. He was elected a member of the prestigious Royal Society of London in 1712. It was a long fight between Brook Taylor and Johann Bernoulli on who discovered and deserves a credit for integration by parts technique.

Taylor theorem

Taylor's Theorem: Let f(x) have n+1 continuous derivatives on interval [⊂, b] ⊂ ℝ for some n ≥ 0, and let x₀ ∈ [⊂, b]. Then \[ f(x) = p_n (x) + R_n (x) , \] where \begin{equation} \label{EqTaylor.1} p_n (x) = \sum_{k=0}^n \frac{(x- x_0 )^k}{k!}\, f^{(k)} (x_0 ) \end{equation} is called Taylor's polynomial of order n and the remainder can be written in two forms (integral and Lagrange): \begin{equation} \label{EqTaylor.2} R_n (x) = \frac{1}{n!} \int_{x_0}^x (x-t)^n f^{(n+1)} (t)\,{\text d}t = \frac{(x - x_0 )^{n+1}}{(n+1)!}\, f^{(n+1)} \left( \xi_n \right) . \end{equation} Here the point ξ is between x and x₀.

The point x₀ is usually chosen at the discretion of the user and is often taken to be 0. Note that the two forms of the remainder are equivalent: the "pointwise" form can be derived from the "integra;" form.    

Example 1: White business example    ■
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Example 2: White business example    ■
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Example 3: White business example    ■
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