Taylor's theorem


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Taylor's theorem

[′tā·lərz ‚thir·əm]
(mathematics)
The theorem that under certain conditions a real or complex function can be represented, in a neighborhood of a point where it is infinitely differentiable, as a power series whose coefficients involve the various order derivatives evaluated at that point.
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Taylor's theorem and Proposition 4.1 show that there are [v.sub.0], ..., [v.sub.N-1] [member of] [B.sub.0] such that
By continuity of [[partial derivative].sup.n]G(x,t)/[partial derivative][x.sup.n] at b and Taylor's theorem one has (2.27).
JACKSON, q-form of Taylor's theorem, Messenger of Math., 38 (1909), pp.
By Taylor's Theorem, there exists [[delta].sub.0] > 0 small enough such that for [absolute value of x] [less than or equal to] [[mu].sup.-1] [[delta].sub.0] we have:
In particular, they explain three principles that they use throughout but that students today may not be familiar with: the square root of minus one, the exponential series and its connection with the binomial theorem, and Taylor's theorem. Among the topics are damped simple harmonic motion, transverse wave motion, waves in more than one dimension, and non-linear oscillation.
By Taylor's theorem, we expand [[PHI].sub.1] (t,r; q), [[PHI].sub.2](t, r; q) in power series of the embedding parameter q as follows:
In such a case, the Taylor's theorem ensures that the remainder term e, that is the approximation error given by the difference between the real value attained by the function and its Taylor polynomial, is negligible if compared to the size of [(x - a).sup.3] when x approaches the value a.
The point is that Lagrange [9], to develop his method, applied Taylor's theorem to the side condition [phi]([x.sub.0] + h,[y.sub.0] + k) = 0 to obtain
Cauchy was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder.
He begins by picking up discrete calculus, including proof by induction, and moves to selected area computation (pi, anyone?), limit's and Taylor's theorem, including series representations and Taylor polynomials, infinite series, including both the positive and the general, beginning logic, including propositional logic, predicates and quantifiers, and proofs, real numbers, functions such as derivatives and a substantial pair of chapters on integrals.
The usual form of Taylor's theorem is then given, leading to the observation that: