泰勒级数

Taylor Series

实质是泰勒公式的项数趋于无穷，对函数的精确表达。利用函数的局部信息，在函数的某一点附近（特定区间）将函数展开为幂级数。

如果 $f (x)$ 在点 $x_{0}$ 具有任意阶导数，则下式称为称为在点 $x_{0}$ 处的泰勒级数：

\begin{array}{r} \sum_{n = 0}^{\infty} \frac{f^{(n)} (x_{0})}{n!} (x - x_{0})^{n} = f (x_{0}) + f^{'} (x_{0}) (x - x_{0}) + \frac{f^{″} (x_{0}) (x - x_{0})}{2!} + \dots + \frac{f^{(n)} (x_{0})}{n!} (x - x_{0}) + \dots \end{array}

通过函数在自变量零点的导数求得的泰勒级数又叫做麦克劳林级数:

\begin{array}{r} \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} = f (0) + f^{'} (0) x + \frac{f^{″} (0) x}{2!} + \dots + \frac{f^{(n)} (0)}{n!} x + \dots \end{array}

\begin{array}{r} e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{1}{2!} x^{2} + \frac{1}{3!} x^{3} + \frac{1}{4!} x^{4} + \dots - \infty < x < + \infty \end{array}

\begin{aligned} s i n (x) & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!} = x - \frac{1}{3!} x^{3} + \frac{1}{5!} x^{5} - \frac{1}{7!} x^{7} + \dots - \infty < x < + \infty \\ \cos (x) & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{(2 n)!} = 1 - \frac{1}{2!} x^{2} + \frac{1}{4!} x^{4} - \frac{1}{6!} x^{6} + \dots - \infty < x < + \infty \end{aligned}

\begin{array}{r} \frac{1}{1 - x} = \sum_{n = 0}^{\infty} x^{n} \Rightarrow \frac{1}{1 + x} = \sum_{n = 0}^{\infty} (- 1)^{n} x^{n} \Rightarrow \frac{1}{1 + x^{2}} = \sum_{n = 0}^{\infty} (- 1)^{n} x^{2 n} - 1 < x \leq 1 \end{array}

\begin{array}{r} \ln (1 + x) = \int \frac{1}{1 + x} d x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{n + 1}}{n + 1} - 1 < x \leq 1 \end{array}

\begin{array}{r} \arctan (x) = \int \frac{1}{1 + x^{2}} d x = \sum_{n = 0}^{\infty} \frac{(- 1)^{n} x^{2 n + 1}}{2 n + 1} - 1 \leq x \leq 1 \end{array}

\begin{array}{r} (1 + x)^{k} = \sum_{n = 0}^{\infty} (\binom{k}{n}) x^{n} = 1 + k x + \frac{k (k - 1)}{2!} x^{2} + \dots + \frac{k (k - 1) \dots (k - n + 1)}{n!} x^{n} + \dots (- 1 < x < 1) \end{array}

\begin{array}{r} f (z) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (z_{0})}{n!} (z - z_{0})^{n} \end{array}

二元泰勒级数展开：

\begin{array}{r} y = f (x_{1}, x_{2}) = f (x_{10}, x_{20}) + [{(\frac{\partial f}{\partial x_{1}})}_{x_{10}, x_{20}} (x_{1} - x_{10}) + {(\frac{\partial f}{\partial x_{2}})}_{x_{10}, x_{20}} (x_{2} - x_{20})] + \dots \end{array}

非线性系统的线性化

微分方程幂级数解法：矩阵指数函数