Taylor Polynomials

Overview

Let $f$ be a function with derivatives of order $n$ at the point $x = a$ . Then there exists exactly one polynomial $P$ of degree $\leq n$ which satisfies the $n + 1$ conditions

P (a) = f (a), P^{'} (a) = f^{'} (a), \dots, P^{(n)} (a) = f^{(n)} (a) .

This polynomial, the Taylor polynomial of degree $n$ generated by $f$ at $a$ , is given by formula

P (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} (x - a)^{k} .

This is often written instead using operator $T_{n}$ on $f$ . That is, $P = T_{n} f$ or $P = T_{n} (f)$ . To indicate the dependence on $a$ , we can instead write $P = T_{n} (f; a)$ .

Linearity

Let $f$ and $g$ be functions with derivatives of order $n$ at point $a$ . Let $c_{1}$ and $c_{2}$ be constants. Then

T_{n} (c_{1} f + c_{2} g; a) = c_{1} T_{n} (f; a) + c_{2} T_{n} (g; a) .

Differentiation

Let $f$ be a function with derivatives of order $n$ at point $a$ . Then

{[T_{n} (f; a)]}^{'} = T_{n - 1} (f^{'}; a) .

Integration

Let $f$ be a function with derivatives of order $n$ at point $a$ . Let $g (x) = \int_{a}^{x} f (t) d t$ . Then

T_{n + 1} g (x; a) = \int_{a}^{x} T_{n} f (t; a) d t .

Substitution

Let $f$ be a function with derivatives of order $n$ at point $a$ . Let $g (x) = f (c x)$ , where $c$ is a constant. Then

T_{n} g (x; a) = T_{n} f (c x; c a) .

Remainder

The remainder (or error) of a Taylor polynomial $T_{n} f$ is the difference

E_{n} (x) = f (x) - T_{n} f (x) .

If $f$ has a continuous derivative of order $n + 1$ at $a$ , then Taylor's formula with remainder $E_{n} (x; a)$ is defined as

f (x) = \sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} (x - a)^{k} + E_{n} (x; a),

where

E_{n} (x; a) = \frac{1}{n!} \int_{a}^{x} (x - t)^{n} f^{(n + 1)} (t) d t .