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The Taylor series of any polynomial is the polynomial itself.. The Maclaurin series of 1 / 1 − x is the geometric series + + + +. So, by substituting x for 1 − x, the Taylor series of 1 / x at a = 1 is
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.
In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms).
Thus to -approximate () = using a polynomial with lowest degree 3, we do so for () with < / by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of g ( x ) {\displaystyle g(x)} , obtaining an approximation of lowest degree 9, 27, 81...
In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the Taylor polynomial (truncated Taylor series) of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions.
A polynomial equation, also called an algebraic equation, is an equation of the form [24] + ... An important example in calculus is Taylor's theorem, ...
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The sine function and all of its Taylor polynomials are odd functions. The cosine function and all of its Taylor polynomials are even functions. In mathematics , an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain .