Taylor's Theorem

Taylor's Theorem approximates a smooth function near a point using a polynomial whose coefficients are determined by the function's derivatives. The remainder term quantifies the error. This is the foundation for numerical methods, asymptotic analysis, and series expansions in analysis.

Statement

Theorem5.1Taylor's Theorem with Lagrange Remainder

Let $f : [a, b] \to \mathbb{R}$ be $n+1$ times differentiable. For $x_0 \in [a, b]$ and $x \in [a, b]$ , there exists $c$ between $x_0$ and $x$ such that

$f(x) = \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k + \frac{f^{(n+1)}(c)}{(n+1)!} (x - x_0)^{n+1}.$

The sum is the Taylor polynomial $T_n(x)$ , and the last term is the Lagrange remainder $R_n(x)$ .

RemarkSpecial cases

$n = 0$ : Mean Value Theorem.
$n = 1$ : Linear approximation $f(x) \approx f(x_0) + f'(x_0)(x - x_0)$ .
$n = 2$ : Quadratic approximation including second derivative.

ExampleTaylor series for e^x

For $f(x) = e^x$ , all derivatives are $f^{(k)}(x) = e^x$ . At $x_0 = 0$ , the Taylor polynomial is

$T_n(x) = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}.$

The remainder satisfies $|R_n(x)| \leq \frac{e^{|x|}|x|^{n+1}}{(n+1)!} \to 0$ as $n \to \infty$ for fixed $x$ , so $e^x = \sum_{k=0}^\infty x^k/k!$ .

ExampleTaylor series for sin(x)

For $f(x) = \sin(x)$ at $x_0 = 0$ , the derivatives cycle: $\sin, \cos, -\sin, -\cos, \ldots$ At $0$ : $0, 1, 0, -1, 0, 1, \ldots$ Thus

$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}.$

Applications

ExampleApproximating functions

To approximate $\sqrt{1 + x}$ near $x = 0$ , use $f(x) = (1 + x)^{1/2}$ . Then $f(0) = 1$ , $f'(0) = 1/2$ , $f''(0) = -1/4$ . The quadratic Taylor approximation is

$\sqrt{1 + x} \approx 1 + \frac{x}{2} - \frac{x^2}{8}.$

For $x = 0.1$ , this gives $\sqrt{1.1} \approx 1.04875$ , very close to the true value $1.04881\ldots$

ExampleComputing limits using Taylor series

$\lim_{x \to 0} \frac{\sin(x) - x}{x^3} = \lim_{x \to 0} \frac{(x - x^3/6 + O(x^5)) - x}{x^3} = \lim_{x \to 0} \frac{-x^3/6 + O(x^5)}{x^3} = -\frac{1}{6}.$

Summary

Taylor's Theorem provides polynomial approximations to smooth functions:

The Taylor polynomial matches the function and its derivatives at a point.
The remainder term estimates the error.
Applications: numerical approximation, asymptotic analysis, evaluating limits.

See Taylor's Theorem proof and Series of Functions for more on convergence.