Remark 2

Tag 0854, permanent. Cite it as [Bourbaki, Tag 0854].

Functions of a Real Variable, chapter I, § 3. DERIVATIVES OF HIGHER ORDER

With the same hypotheses, the coefficients $a_i$ in (11) are determined recursively by the relations

a_0 = f(a)

a_1 = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}

a_2 = \lim_{x \to a} \frac{f(x) - f(a) - a_1(x - a)}{(x - a)^2}

\ldots

a_n = \lim_{x \to a} \frac{f(x) - f(a) - a_1(x - a) - \cdots - a_{n-1}(x - a)^{n-1}}{(x - a)^n}.

In the case $a = 0$ one concludes, in particular, that if $f(x^p)$ ( $p$ an integer > 0) has a derivative of order $pn$ on a neighbourhood of 0 then the Taylor expansion of order $pn$ of this function is simply

f(x^p) = a_0 + a_1 x^p + a_2 x^{2p} + \cdots + a_n x^{np} + r_n(x^p)

where $r_n(x^p)$ is the remainder in the expansion (cf. V, p. 222).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

V, p. 222