Proposition 3

Tag 0851, permanent. Cite it as [Bourbaki, Tag 0851].

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

If $\|f^{(n+1)}(x)\| \leq M$ on I, then we have

\|r_n(x)\| \leq M \frac{|x - a|^{n+1}}{(n + 1)!}

on I.

Indeed, the formula holds for $n = 0$ , by I, p. 15, th. 2. Let us prove it by induction on $n$ : by the induction hypothesis applied to $f'$ , one has

\|r'_n(y)\| \leq M \frac{|y - a|^n}{n!}

from which the formula (9) follows by the mean value theorem (I, p. 23, th. 2).

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

Cited by

Nothing.

Cites

I, p. 15