The remainder — Lagrange, Peano, Cauchy

The Taylor polynomial approximates $f$, but with what error? The remainder $R_n(x) = f(x) - T_n(x)$ can be written in three equivalent forms, each useful in a different context.

Theorem — Taylor's formula with Lagrange remainder

If $f$ is differentiable $n+1$ times in a neighbourhood of $x_0$, then for every $x$ in that neighbourhood there exists a point $c$ strictly between $x_0$ and $x$ such that $f(x) = T_n(x) + \underbrace{\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}}_{R_n(x):\ \text{Lagrange remainder}}.$

Remark — A linguistic reading: " $\forall x\,\exists c$"

The logical structure of the formula is nested: for every $x$ in the neighbourhood, there exists a $c$ (which depends on $x$!) such that the equality holds. It is a "$\forall x\,\exists c$", not an "$\exists c\,\forall x$": the point $c$ changes with $x$ and is not known exactly, one knows only that it lies between $x_0$ and $x$. For numerical applications this is enough: to bound the error $|R_n(x)|$ one bounds $|f^{(n+1)}|$ over the whole interval (so the estimate holds whatever $c$ may be). Being able to read "$\forall\ldots\exists\ldots$" is, as Esty observes, a skill of a higher level than mere algebraic manipulation.

Theorem — Taylor's formula with Peano remainder

If $f$ is differentiable $n$ times at $x_0$, then as $x\to x_0$ $f(x) = T_n(x) + o\bigl((x-x_0)^n\bigr),$ where $o\bigl((x-x_0)^n\bigr)$ is a function that tends to zero more rapidly than $(x-x_0)^n$ (little-o).

Theorem — Taylor's formula with Cauchy remainder

Under Lagrange’s hypotheses, there exists $c$ between $x_0$ and $x$ such that $R_n(x) = \frac{f^{(n+1)}(c)}{n!}(x-c)^n(x-x_0).$

Remark — When to use which

• Lagrange: needed when one wants to estimate numerically the approximation error (it suffices to bound $|f^{(n+1)}|$ over the interval).
• Peano: needed when one wants to compute an indeterminate limit by replacing $f$ with its Taylor series.
• Cauchy: an alternative form of Lagrange, useful in some technical proof (rarely in calculations).

Links

Topics: Derivatives
Concepts: Little-o · Taylor polynomial · Taylor remainder
Methods: Taylor Lagrange remainder · Taylor Peano remainder
Skills: Estimating
People: Augustin-Louis Cauchy · Joseph-Louis Lagrange · Peano