Theorem 1

If the function $f$ has an $n^{th}$ derivative at the point $a$ then

We proceed by induction on $n$. The theorem holds for $n = 1$. For arbitrary $n$ one can, by the induction hypothesis, apply it to the derivative $f'$ of $f$: for any $\varepsilon > 0$ there is an $h > 0$ such that, if one puts

one has, for $|y-a| \leq h$ and $y \in I$,

We apply the mean value theorem (I, p. 15, th. 2) on the interval with endpoints $a,\ x$ (with $|x-a| \leq h$) to the vector function $g$ and to the real increasing function equal to $\varepsilon |y - a|^n / n$ if $x > a$, and to $-\varepsilon |y - a|^n / n$ if $x < a$; it follows that $\|g(x)\| \leq \varepsilon |x - a|^n / n$, which proves the theorem.

We thus can write

where $u(x)$ approaches 0 as $x$ approaches $a$ while remaining in I; this formula is called Taylor’s formula of order $n$ at the point $a$, and the right-hand side of (8) is called the Taylor expansion of order $n$ of the function $f$ at the point $a$. The last term $r_n(x) = u(x)(x - a)^n / n!$ is called the remainder in the Taylor formula of order $n$.

When $f$ has a derivative of order $n + 1$ on I, one can estimate $\|r_n(x)\|$ in terms of this $n + 1^{th}$ derivative, on all of I, and not just on an unspecified neighbourhood of $a$: