> [!definition] > > Let $f$ be a [[Function|function]] that is infinitely [[Derivative|differentiable]] at $a$, then the Taylor Series of $f$ at $a$ is a power series based on its [[Derivative|derivatives]]. > $ > \sum_{n = 0}^{\infty}\frac{\frac{d^n}{dx^n}(a)}{n!}(x - a)^{n} > $ > > If $a = 0$, the series is known as a Maclaurin series. > > Let $k \ge 0$, > $ > T_k(x) = \sum_{n = 0}^{k}\frac{\frac{d^n}{dx^n}(a)}{n!}(x - a)^{n} > $ > is known as the *degree $k$ Taylor polynomial of $f$ at $a$*. > [!note] Taylor's Theorem > > Let $k \ge 1$ be an integer and let the function $f$ be $k$ times [[Derivative|differentiable]] at point $a \in \real$. Then there exists a function $h_k$ such that > $ > f(x) = \sum_{n = 0}^{k}\frac{\frac{d}{dx}(a)}{n!}(x - a)^n + h_k(x) > $ > And $\lim_{x \to a}h_k(x) = 0$ (error approaches $0$) close to the approximation point. > > Meaning that as $k \to \infty$, the Taylor series becomes equal to the original function. > [!note] Taylor's Inequality > > If $\frac{d^{k + 1}f}{dx^{k + 1}}$ is bounded by some constant $M$ on some interval $(a - R, a + R)$. Then the remainder $|R_k(x)| \le \frac{M}{(k + 1)!}|x - a|^{k + 1}$.