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Since a second-order expansion for [()] has already been derived above, it only remains to find [() ()]. Treating () as a two-variable function, the second-order Taylor expansion is as follows:
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, [2] although an earlier version of the result was already mentioned in 1671 by James Gregory. [3] Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis.
In fact, for a smooth enough function, we have the similar Taylor expansion (+) = | | ()! + (,), where the last term (the remainder) depends on the exact version of Taylor's formula.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
Created Date: 8/30/2012 4:52:52 PM
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem.Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at the points of A.
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...