Taylor and Maclaurin Series

Taylor series provide the canonical power series representation of a smooth function, where the coefficients are determined by the function's derivatives at a single point.

Taylor Series

Definition

If $f$ is infinitely differentiable at $x = a$ , the Taylor series of $f$ centered at $a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$ When $a = 0$ , this is called the Maclaurin series: $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$

The Taylor polynomial $T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k$ is the unique polynomial of degree $\leq n$ that matches $f$ and its first $n$ derivatives at $x = a$ .

Definition

The Taylor remainder (or error term) is $R_n(x) = f(x) - T_n(x)$ . The Taylor series of $f$ converges to $f(x)$ if and only if $R_n(x) \to 0$ as $n \to \infty$ . The Lagrange form of the remainder is $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}$ for some $c$ between $a$ and $x$ .

Essential Maclaurin Series

ExampleFundamental Maclaurin series

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \quad R = \infty$ $\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \quad R = \infty$ $\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}, \quad R = \infty$ $\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n}, \quad R = 1$ $(1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n, \quad R = 1 \text{ (for non-integer } \alpha\text{)}$ $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \quad R = 1$

Convergence to the Function

RemarkNot all smooth functions equal their Taylor series

There exist $C^\infty$ functions whose Taylor series converges everywhere but does not equal the function. The classic example is $f(x) = e^{-1/x^2}$ for $x \neq 0$ , $f(0) = 0$ . One can show $f^{(n)}(0) = 0$ for all $n$ , so the Maclaurin series is identically $0$ , yet $f(x) > 0$ for $x \neq 0$ . Functions equal to their Taylor series are called analytic (or real-analytic).

Taylor series provide the systematic bridge between differential calculus and the algebra of power series, enabling computations in analysis, physics, and numerical methods.