A sufficiently differentiable function can be approximated near a point by a polynomial built from its derivatives at that point, the Taylor polynomial, with a remainder term that can be bounded explicitly. Named for Brook Taylor, it is the basis for power series expansions throughout analysis.
Facts
StatementTaylor's theorem states that a k-times differentiable function can be approximated near a given point by a polynomial of degree k, called the k-th order Taylor polynomial, built from the function's own derivatives at that point. 1 Classification
Statement Form Connections
Sources
1. Taylor's Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial.
History section
Taylor's theorem is named after Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.
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