---
tag: 005Q84BMAD
kind: statement
role: theorem
name: Taylor's theorem with the Cauchy remainder
msc: [41A80]
uses: [0HFSFVMWQB, 0BARQY52N5]
sources:
  - url: https://en.wikipedia.org/wiki/Taylor%27s_theorem
    locator: Explicit formulas for the remainder
---

## Statement

Let $k \ge 0$ and let $a \ne x$ be real numbers. Let $J$ be the closed interval with endpoints $a$ and $x$.
Let $f$ be a real function with $f^{(k)}$ continuous on $J$ and $f^{(k+1)}$ existing on the interior of $J$.
Then there is $\xi$ strictly between $a$ and $x$ with
$$f(x) = P_k(x) + \frac{f^{(k+1)}(\xi)}{k!} (x - \xi)^k (x - a),$$
where $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.

## Remarks

Apply the [[mean value theorem|0BARQY52N5]] to $F(t) = f(t) + \sum_{j=1}^{k} f^{(j)}(t) (x - t)^j / j!$ on $J$. Note that $F(x) = f(x)$ and $F(a) = P_k(x)$.
This form sometimes gives better bounds than the [[Lagrange form|0H5ZAZCX2T]], for example for $\ln(1 + x)$ with $-1 < x < 0$.
Mathlib states it as `taylor_mean_remainder_cauchy`.

## Explanations

### L1 human

The Cauchy form is another exact expression for the error.
It moves part of the power of $(x - a)$ to a power of $(x - \xi)$.