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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.
Proof: This can be shown ... This parallels the reasoning behind the Lagrange remainder term in the Taylor theorem; in fact, the Taylor remainder is a special case of ...
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 .
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.
The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder is not guaranteed. [8] Polynomial division leads to a result known as the polynomial remainder theorem: If a polynomial f(x) is divided by x − k, the remainder is the constant r = f(k). [9] [10]
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: () + ′ ().
Taylor Swift has announced a new edition of forthcoming album The Tortured Poets Department, featuring a bonus track called “The Bolter.”Almost immediately after Swift dropped the news at a ...
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)