Taylor's Theorem¶

Theorem¶

Let $k \geq 1$ be an integer and assume that the function $f\colon \mathbb{R}\to\mathbb{R}$ is $k$ -times differentiable (not necessarily real analytic) at $x=a\text{.}$ There exists a function $\displaystyle h_k \colon \mathbb{R} \to\mathbb{R}$ such that $\displaystyle \lim_{x\to a} h_k(x) = 0$ and

f(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(k)}(a)}{k !}(x-a)^k + R_k(x)

where $R_k(x)= h_k(x)(x-a)^k$ is called the Peano form of the remainder.

Interpretation¶

Taylor's theorem says that a function which is $k$ times differentiable at $a$ is, near $a$ , a polynomial plus a small error. The polynomial is the Taylor polynomial of order $k$ , and the error $R_k(x)=h_k(x)(x-a)^k$ is small even compared with the last term kept, because the factor $h_k(x)$ tends to $0$ as $x\to a$ $a$ .

The exponential curve and its second-degree Taylor polynomial 1 + x + x squared over 2 around 0; at a point x the vertical gap between them is the remainder.

The figure takes $f(x)=\mathrm{e}^x$ and $a=0$ $0$ , with the second-order Taylor polynomial $P_2(x)=1+x+\tfrac{x^2}{2}$ $\frac{x ^{2}}{2}$ . The two curves touch at $0$ ; at the marked point $x$ the green arrow is the remainder $R_2(x)=f(x)-P_2(x)$ $P_{2} (x)$ . As $x$ moves towards $0$ this gap shrinks faster than $x^2$ , which is the statement $h_2(x)=R_2(x)/x^2\to0$ $0$ .

The right panel shows $h_k(x)=R_k(x)/(x-a)^k$ $a)^{k}$ : it tends to $0$ as $x\to a$ $a$ , which is the Peano form of the remainder.

The hypothesis is weak: only $f^{(k)}(a)$ has to exist, and $f$ need not be smooth or real analytic. In return, the conclusion only concerns the limit $x\to a$ $a$ , not the error at a fixed $x$ . The order of the expansion is limited by the number of derivatives: $f(x)=x|x|$ has $f'(0)=0$ and $f(x)=0+0\cdot x+x|x|$ with $h_1(x)=|x|\to0$ $0$ , but $f''(0)$ does not exist and $x|x|$ has no expansion $c_0+c_1x+c_2x^2+o(x^2)$ $o (x^{2})$ , since $x|x|/x^2=\operatorname{sgn}(x)$ has no limit at $0$ .

In little- $o$ notation, the theorem turns limits of quotients into a comparison of leading terms, an alternative to L'Hôpital's rule: from $\sin x=x-\tfrac{x^3}{6}+o(x^3)$ one reads off $(x-\sin x)/x^3\to\tfrac{1}{6}$ as $x\to0$ $0$ .

The theorem returns for functions of several variables in Math IV, in multivariate Taylor approximations.

Key Results¶

Using the little- $o$ notation, the statement in Taylor's theorem reads as $R_k(x)=o((x-a)^k)\text{,}$ as $x\to a\text{.}$ In this case, we have the $k$ -th order Taylor expansion, whose polynomial part is the Taylor polynomial $P_k(x)$ :

f(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(k)}(a)}{k !}(x-a)^{k} + o((x-a)^{k}).

Examples¶

$\mathrm{e}^x=1+x+\dfrac{x^2}{2}+h_2(x)x^2$ $h_{2} (x) x^{2}$ , where $\displaystyle h_2(x)=\sum_{n=3}^\infty \dfrac{x^{n-2}}{n!}$ $n = 3 \sum \infty \frac{x ^{n - 2}}{n !}$ .
$\dfrac{1}{1-x}=1+x+x^2+h_2(x) x^2$ $h_{2} (x) x^{2}$ , where $\displaystyle h_2(x)=\sum_{n=3}^\infty x^{n-2}$ $n = 3 \sum \infty x^{n - 2}$ , for $|x|<1$ $1$ .

Prerequisites¶

Higher-Order Derivative

Sources¶

Math III: Analysis · Lecture 3

Aliases: Taylor's formula, Taylor’s formula Keywords: k-th order Taylor expansion, remainder term Symbols: infinity, real numbers, sum

Last updated: 22 October 2026 19:04