Exercise 17, § 1

Functions of a Real Variable, II, p. 32

Let g be a numerical function admitting a continuous derivative and ≠ 0 in [a, x]; if f is a numerical function having a regulated (n + 1)-th derivative in [a, x], now prove that the remainder $r_n(x)$ of the Taylor development of order n of f at the point a, can be written

r_n(x) = (g(x) - g(a)) \frac{(x - \xi)^n}{n!} \frac{f^{(n+1)}(\xi)}{g'(\xi)}

where

a < \xi < x

(use the integral form of

r_n(x)

Solution

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