A differentiable function is known to possess derivatives of all orders on an open…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
A differentiable function p is known to possess derivatives of all orders on an open interval containing the point x=a. On that interval, p can be represented by a Taylor series about x=a.
(a) Derive a general expression that relates the coefficient of (x−a)n in the Taylor series of p to the nth derivative of p evaluated at x=a. Clearly state any assumptions you make in your reasoning.
(b) Describe how your result in part (a) guarantees the uniqueness of every higher–order derivative of p at x=a when the Taylor series is known.
(c) Suppose the coefficient of (x−a)5 in the Taylor series of p is zero. Explain, without performing any explicit differentiation, what this implies about p(5)(a) and about the presence (or absence) of a potential point of inflection at x=a. Justify your answer thoroughly.