Taylor Polynomial¶

Definition¶

Let $k\geq0$ be an integer and let the function $f$ be $k$ times differentiable at $x=a$ $a$ . The $k$ -th order Taylor polynomial of $f$ at $x=a$ is the polynomial

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

Interpretation¶

The $k$ -th order Taylor polynomial is the polynomial of degree at most $k$ that imitates $f$ as closely as possible near $x=a$ $a$ . Its coefficients are chosen so that $P_k$ has the same value and the same first $k$ derivatives at $a$ as $f$ : differentiating $(x-a)^j$ exactly $j$ times leaves the constant $j!$ , which the factorial in the coefficient $f^{(j)}(a)/j!$ cancels.

Read term by term, $P_0$ is the constant $f(a)$ , and $P_1(x)=f(a)+f'(a)(x-a)$ is the tangent line, whose slope is the derivative $f'(a)$ . Each further term corrects what the previous ones miss, starting with the curvature. If $f$ is $k$ times differentiable at $a$ , then $P_k$ is the unique polynomial of degree at most $k$ with $f(x)-P_k(x)=o\big((x-a)^k\big)$ as $x\to a$ (in little- $o$ notation); in this sense it is the best polynomial approximation of order $k$ near $x=a$ $a$ .

The $k$ -th order Taylor polynomial consists of the first $k+1$ terms of the Taylor series centred at the same point, when that series exists.

A common slip is to expect that a higher order always gives a better approximation at a fixed $x$ far from $a$ . The guarantee is local. For $f(x)=1/x$ at $a=1$ $1$ , $\displaystyle P_k(x)=\sum_{j=0}^{k}(-1)^j(x-1)^j$ $1)^{j}$ , so $P_k(3)=1-2+4-\cdots+(-2)^k$ moves further away from $f(3)=\tfrac{1}{3}$ as $k$ grows.

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

Key Results¶

The $k$ -th order Taylor polynomial and the function $f$ have the same value and the same derivatives up to order $k$ at $x=a\text{:}$ $P_k^{(j)}(a)=f^{(j)}(a)$ for $j=0,1,\dots,k\text{.}$ Away from $a$ they generally differ.
One can write a Taylor polynomial for a function just from its first few derivatives, but the full Taylor series exists if the function is smooth, and the full series converges to the function on a neighbourhood of $a$ if and only if the function is real analytic at $a$ .

Examples¶

The 3rd order Taylor polynomial of $\sin x$ around $x=0$ is $P_3(x) = x - \dfrac{x^3}{3!}$ $\frac{x ^{3}}{3 !}$ .

The two panels compare Taylor polynomials of $\sin x$ about $0$ . The cubic $P_3$ on the right follows $\sin x$ over a wider interval than the linear polynomial $P_1$ on the left.
The 1st order Taylor polynomial of $1/x$ around $x = 1$ is $P_1(x)=2-x$ $x$ .

Raise the order $k$ and move the centre $a$ : $P_k$ matches $f$ and its first $k$ derivatives at $a$ , and fits over a wider range.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 3

Symbols: sum

Last updated: 22 October 2026 19:04