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.

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:

formulated using

Taylor's theorem and some conditions

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.

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: