# 微积分笔记（13）——带 Peano 余项的 Taylor 公式

## 微积分笔记（13）——带 Peano 余项的 Taylor 公式

Contents

### Taylor 公式——带 Peano 余项（续）

#### Taylor 公式证明

$$\lim_{\Delta x \to 0} \dfrac{f(x_0 + \Delta x) - P_n(\Delta x)}{\Delta x^n} \\ = \lim_{\Delta x \to 0} \dfrac{f^{\prime}(x_0 + \Delta x) - P_n^{\prime}(\Delta x)}{n\Delta x^{n-1}} \\ \vdots \\ = \lim_{\Delta x \to 0} \dfrac{f^{(n-1)}(x_0 + \Delta x) - f^{n-1}(x_0) - f^{n}(x_0) \Delta x}{n! \Delta x} \\ = \dfrac{1}{n!} \lim_{\Delta x \to 0} [\dfrac{f^{(n-1)}(x_0 + \Delta x) - f^{n-1}(x_0)}{\Delta x} -f^{(n)}(x_0)] \\ = 0 \ \square$$

#### Maclaurin 展开（$x_0 = 0$）

$$f(x) = f(0) + \dfrac{f^{\prime}(0)}{1!} x + \dfrac{f^{\prime \prime}(0)}{2!} x^2 + \cdots + \dfrac{f^{(n)}(0)}{n!} x^n + o(x^n)$$

#### 应用

$$e^x = 1 + \dfrac{x}{1!} + \dfrac{x^2}{2!} + \cdots + \dfrac{x^n}{n!} + o(x^n) \\ \sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots + (-1)^m \dfrac{x^{2m + 1}}{(2m + 1)!} + o(x^{2m + 2}) \\ \cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots + (-1)^m \dfrac{x^{2m}}{(2m)!} + o(x^{2m + 1}) \\ \dfrac{1}{1 + x} = 1 - x + x^2 - x^3 + \cdots +(-1)^n x^n + o(x^n) \\ \ln (1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots + (-1)^{n-1} \dfrac{x^n}{n} + o(x^n) \\ (1 + x)^a = 1 + ax + \dfrac{a(a - 1)}{2!} x^2 + \cdots + \dfrac{a(a - 1) \cdots (a - n + 1)}{n!} x^n + o(x^n)$$

#### 问题

$$P_n(x - x_0) - Q_n(x - x_0) = o((x-x_0)^n) \\ \Rightarrow P_n(x - x_0) = Q_n(x - x_0) \ \square$$

#### 间接展开

$$e^{x^2 + 2x} = e^{(x + 1)^2 - 1} \\ = e^{-1}(1 + \dfrac{(x + 1)^2}{1!} + \dfrac{(x + 1)^4}{2!} + \cdots + \dfrac{(x+1)^{2n}}{n!} + o((x+1)^{2n})) \\ \dfrac{1}{x^2 + 3x} = \dfrac{1}{3}(\dfrac{1}{x} - \dfrac{1}{x + 3}) \\ = \dfrac{1}{3}((\sum_{k = 0}^n (-1)^k (x-1)^k + o((x-1)^n))-(\dfrac{1}{4}\sum_{k = 0}^n (-1)^k (\dfrac{x-1}{4})^k + o((\dfrac{x-1}{4})^n))) \\ = \dfrac{1}{3} \sum_{k = 0}^n (-1)^k (1 - \dfrac{1}{4^{k+1}})(x - 1)^k + o((x - 1)^n) \\ \tan x = \dfrac{\sin x}{\cos x} \\ = \dfrac{x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + o(x^6)}{1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + o(x^5)} \\ = x + \dfrac{x^3}{3} + \dfrac{2}{15} x^5 + o(x^5)$$