Taylor series (and Maclaurin series)

The infinite power series $\sum\dfrac{f^{(n)}(a)}{n!}(x-a)^n$ built from ALL of $f$ 's derivatives; a Maclaurin series is the special case $a=0$ .

The Taylor series of $f$ centered at $a$ is $\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$ — the Taylor polynomial extended to infinitely many terms. A Maclaurin series is simply a Taylor series centered at $a=0$ . The Taylor series converges to $f(x)$ itself (not just approximates it) exactly where the Lagrange remainder shrinks to $0$ .

Examples

The Maclaurin series for $e^x$ is $\sum_{n=0}^{\infty}\dfrac{x^n}{n!}=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots$ , converging to $e^x$ for every real $x$ .