---
tag: 03AJHX431A
kind: statement
role: corollary
name: Taylor's theorem with the Peano remainder
msc: [41A58]
uses: [03QHBVW4TT, 04TSHB9H8D]
sources:
  - url: https://en.wikipedia.org/wiki/Taylor%27s_theorem
    locator: Statement of the theorem
---

## Statement

Let $k \ge 1$, let $I$ be an open interval, let $a \in I$, and let $f \colon I \to \mathbb{R}$ be $k$ times differentiable at $a$.
Then
$$f(x) = \sum_{j=0}^{k} \frac{f^{(j)}(a)}{j!} (x - a)^j + o\big((x - a)^k\big) \quad \text{as } x \to a.$$

## Remarks

This is [[Taylor's theorem|03QHBVW4TT]] written with [[little o notation|04TSHB9H8D]].
For example, $\cos x = 1 - x^2/2 + o(x^3)$ as $x \to 0$.
Mathlib states a form for $C^k$ functions on a convex set as `taylor_isLittleO`.

## Explanations

### L1 human

The Peano form only says that the error is small compared with $(x - a)^k$.
It gives no bound for a fixed $x$, but it needs weak hypotheses.