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In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is a further generalization where only derivatives of some orders are prescribed, not necessarily all orders from 0 to a k .
Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be convergent at all. In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions .
For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as (+) = + ′ ()! + ()! + + ()! + (),. Where 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.
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: () + ′ ().
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
Some Taylor Swift fans found proof Harry Styles may have inspired The Tortured Poets Department track "But Daddy I Love Him." Their theory, explained.
Menger's theorem (graph theory) Milliken–Taylor theorem (Ramsey theory) Milliken's tree theorem (Ramsey theory) Multinomial theorem (algebra, combinatorics) Mycielski's theorem (graph theory) Nicomachus's theorem (number theory) Ore's theorem (graph theory) Paley's theorem ; Perfect graph theorem (graph theory) Perlis theorem (graph theory)