Definition
Taylor Series for One Variable
Let be an infinitely differentiable function at the point , then
Taylor Series for Two Variables
Let be an infinitely differentiable function at the point , then
&+ \frac{\partial f}{\partial t}(t_{0}, x_{0})(t-t_{0}) + \frac{\partial f}{\partial x}(t_{0}, x_{0})(x-x_{0})\\ &+ \frac{1}{2}\frac{\partial^{2} f}{\partial t^{2}}(t_{0}, x_{0})(t-t_{0})^{2} + \frac{1}{2}\frac{\partial^{2} f}{\partial x^{2}}(t_{0}, x_{0})(x-x_{0})^{2} + \frac{\partial^{2} f}{\partial t \partial x}(t_{0}, x_{0})(x-x_{0})(t-t_{0})\\ &+ \dots \end{aligned}$$ ## Approximation Let $f: \mathbb{R} \to \mathbb{R}$ be a $k$-the differentiable function at the point $c \in \mathbb{R}$ and $|x-c| \to 0$, then $$f(x) = \sum\limits_{n=0}^{k} \cfrac{f^{(n)}(c)}{n!}(x-c)^{n} + o(|x-c|^{k})$$ where $o$ is [[Big O Notation|little oh]] notation Let $f: \mathbb{R} \to \mathbb{R}$ be a $k+1$-the differentiable function at any point and $|f^{(k+1)}(x)| \leq M \leq \infty$, then by the [[Rolle's Theorem]] $$f(x) = \sum\limits_{n=0}^{k} \cfrac{f^{(n)}(c)}{n!}(x-c)^{n} + \frac{1}{(k+1)!}f^{(k+1)}(\xi)(x-c)^{k+1}$$ where $\xi \in (x, c)$ ## Maclaurin Series $$f(x) = \sum\limits_{n=0}^{\infty} \cfrac{f^{(n)}(0)}{n!}x^{n} = f(0)+f'(0)x + \cfrac{f''(0)}{2!}x^{2}+ \dots +\cfrac{f^{(m)}(0)}{m!}x^m + \dots$$ Taylor Series with $c=0$ ## Derivation from the [[Fundamental Theorem of Calculus]] Let $f$ be an infinitely [[Differentiability|differentiable]] function. By applying [[Fundamental Theorem of Calculus|FTC]] $n$ times, we can expande the function $f(x)$ as follows: $$\begin{aligned} f(x) &= \int\limits_{c}^{x} f'(x_{1}) dx_{1} + f(c)\\ &= \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} f''(x_{2}) dx_{2} + f'(c) dx_{1} + f(c)\\ &= \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} f''(x_{2}) dx_{2} dx_{1} + \int\limits_{c}^{x} f'(c) dx_{1} + f(c)\\ &= \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\int\limits_{c}^{x_{2}} f'''(x_{3}) dx_{3} dx_{2} dx_{1} + \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} f''(c) dx_{2} dx_{1} + \int\limits_{c}^{x} f'(c) dx_{1} + f(c)\\ &= \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{n-1}} f^{(n)}(x_{n}) dx_{n} dx_{n-1} \dots dx_{1} + \sum_{i=0}^{n} \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{i-1}} f^{(i)}(c) dx_{i} dx_{i-1} \dots dx_{1} \end{aligned}$$ The integrals inside the summation, can be simplified $$\begin{aligned} \int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{i-1}} f^{(i)}(c) dx_{i} dx_{i-1} \dots dx_{1} &= f^{(i)}(c)\int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{i-1}} 1 dx_{i} dx_{i-1} \dots dx_{1} \\ &= f^{(i)}(c)\int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{i-2}} (x_{i-1} - c) dx_{i-1} dx_{i-2} \dots dx_{1} \\ &= f^{(i)}(c)\int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{i-3}} \frac{1}{2}((x_{i-2} - c)^2 - (c - c)^2) dx_{i-2} dx_{i-3} \dots dx_{1} \\ &\dots \\ &= f^{(i)}(c)\frac{(x - c)^i}{i!} \end{aligned}$$ Therefore, we have $$f(x) = \underbrace{\int\limits_{c}^{x} \!\int\limits_{c}^{x_{1}} \!\dots \!\int\limits_{c}^{x_{n-1}} f^{(n)}(x_{n}) dx_{n} dx_{n-1} \dots dx_{1}}_{\text{Error term}} + \underbrace{\sum_{i=0}^{n} f^{(i)}(c)\frac{(x - c)^i}{i!}}_{\text{Polynomial terms}}$$