泰勒公式与等价无穷小

(泰勒定理/泰勒公式)任何n阶可导函数都可以写成f(x)=∑i=1nanxnf(x)=\sum\limits_{i=1}^n a_nx^n. ⭐️⭐️⭐️

  1. 带拉格朗日余项的泰勒公式(用于证明):
    f(x)n+1f(x)n+1阶可导 ⇒\Rightarrow
    f(x)=f(x0)+f′(x0)(x−x0)+f′′(x0)2!(x−x0)2+⋯+f(n)(x0)n!(x−x0)n+f(n+1)(ξ)(n+1)!(x−x0)n+1f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}

    其中f(n)(x0)n!(x−x0)n\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n为通项,f(n+1)(ξ)(n+1)!(x−x0)n+1\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}为拉氏余项,ξ\xi介于xx与x0x_0之间.

  2. 带佩亚诺余项的泰勒公式(用于计算):
    若f(x)nf(x)n阶可导 ⇒\Rightarrow
    f(x)=f(x0)+f′(x0)+f′′(x0)2!(x−x0)2+⋯+f(n)(x0)n!(x−x0)n+o((x−x0)n)f(x)=f(x_0)+f'(x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 +\cdots+ \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o((x-x_0)^n)

    其中o((x−x0)n)o((x-x_0)^n)为佩亚诺余项,表示(x−x0)n(x-x_0)^n的高阶无穷小.

  3. 当x0=0x_0=0时,泰勒公式成为麦克劳林公式
    f(x)=f(0)+f′(0)x+f′′(0)2!x2+⋯+f(n)(0)n!xn+o(xn)f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\cdots+ \frac{f^{(n)}(0)}{n!}x^n+o(x^n)

x→0x\rightarrow 0时的常用泰勒公式:

sin⁡x=x−x33!+x55!+o(x5)\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!}+ o(x^5)

arcsin⁡x=x+12x33+1234x55+o(x3)\arcsin x = x + \frac{1}{2}\frac{x^3}{3} + \frac{1}{2}\frac{3}{4}\frac{x^5}{5} + o(x^3)

cos⁡x=1−x22!+x44!+o(x4)\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + o(x^4)

tan⁡x=x+13x3+215x5+o(x5)\tan x = x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + o(x^5)

arctan⁡x=x−x33+x55+o(x5)\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} + o(x^5)

ln⁡(1+x)=x−x22+x33−x44+o(x4)\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + o(x^4)

ex=1+x+x22!+x33!+o(x3)e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + o(x^3) 🌠

(1+x)a=1+ax+a(a−1)2!x2+a(a−1)(a−2)3!x3+o(x3)(1+x)^a = 1 + ax + \frac{a(a-1)}{2!}x^2 + \frac{a(a-1)(a-2)}{3!}x^3+o(x^3)

11−x=1+x+x2+x3+o(x3)\frac{1}{1-x} = 1 + x + x^2 + x^3 + o(x^3)

11+x=1−x+x2−x3+o(x3)\frac{1}{1+x} = 1 - x + x^2 - x^3 + o(x^3)

1+x=1+x2−x28+o(x2)\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + o(x^2)

11+x=1−x2+38x3+o(x3)\frac{1}{\sqrt{1+x}} = 1 - \frac{x}{2}+\frac{3}{8}x^3+o(x^3)


等价无穷小

等价无穷小由泰勒公式推导而来, 可以通过泰勒公式的组合推导出更多的等价无穷小

  1. sin⁡x∼x\sin x \sim x
  2. arcsin⁡x∼x\arcsin x \sim x
  3. tan⁡x∼x\tan x \sim x
  4. arctan⁡x∼x\arctan x \sim x
  5. ex−1∼xe^x-1 \sim x
  6. ln(1+x)∼xln(1+x) \sim x
  7. (1+x)α−1∼αx(1+x)^\alpha-1 \sim \alpha x
  8. 1−cos⁡x∼12x21-\cos x \sim \frac{1}{2}x^2
  9. x−sin⁡x∼16x3x-\sin x \sim \frac{1}{6}x^3
  10. x+sin⁡x=2xx + \sin x = 2x
  11. x2−sin⁡2x=x43x^2 - \sin^2 x = \frac{x^4}{3}
  12. x−tan⁡x=−x3xx-\tan x=-\frac{x^3}{x}
  13. x→1,lnx=ln(1+x−1)∼x−1x\rightarrow 1, lnx=ln(1+x-1) \sim x-1 🌠🌠

扩展等价无穷小的用法

(x→0)(x\rightarrow 0)

  1. ex−1∼xe^x-1 \sim x
    当f(x)→0,g(x)→0)f(x)\rightarrow 0, g(x)\rightarrow 0)

    1. ef(x)−1∼f(x)e^{f(x)}-1 \sim f(x)
    2. ef(x)−eg(x)=eg(x)(ef(x)−g(x)−1)∼eg(x)(f(x)−g(x))∼k(f(x)−g(x))e^{f(x)} - e^{g(x)} = e^{g(x)}(e^{f(x)-g(x)}-1) \sim e^{g(x)}(f(x)-g(x)) \sim k(f(x)-g(x))

    eg1: lim⁡x→0e−ecos⁡x1+x23−1=lim⁡x→0ecos⁡x(e1−cos⁡x−1)13x2=lim⁡x→03e(1−cos⁡x)x2=lim⁡x→03e12x2x2=32e\lim\limits_{x\rightarrow 0}\frac{e - e^{\cos x}}{\sqrt[3]{1+x^2}-1}=\lim\limits_{x\rightarrow 0}\frac{e^{\cos x}(e^{1-\cos x}-1)}{\frac{1}{3}x^2}=\lim\limits_{x\rightarrow 0}3e\frac{(1-\cos x)}{x^2}=\lim\limits_{x\rightarrow 0}3e\frac{\frac{1}{2}x^2}{x^2}=\frac{3}{2}e
    eg2: lim⁡x→0ex−esin⁡xx3=lim⁡x→0esin⁡x(ex−sin⁡x−1)x3=lim⁡x→01⋅(x−sin⁡x)x3=16\lim\limits_{x\rightarrow 0}\frac{e^x-e^{\sin x}}{x^3}=\lim\limits_{x\rightarrow 0}\frac{e^{\sin x}(e^{x-\sin x} - 1)}{x^3}=\lim\limits_{x\rightarrow 0}\frac{1\cdot(x-\sin x)}{x^3}=\frac{1}{6}

  2. ln⁡(1+x)∼x\ln(1+x) \sim x
    当f(x)→1f(x)\rightarrow 1

    1. ln⁡f(x)=ln⁡(1+f(x)−1)∼f(x)−1\ln f(x) = \ln(1+f(x)-1) \sim f(x)-1

    eg1: lim⁡x→0ln⁡cos⁡xx2=lim⁡x→0cos⁡x−1x2=lim⁡x→0−12x2x2=−12\lim\limits_{x\rightarrow 0}\frac{\ln\cos x}{x^2} = \lim\limits_{x\rightarrow 0}\frac{\cos x-1}{x^2}=\lim\limits_{x\rightarrow 0}\frac{-\frac{1}{2}x^2}{x^2}=-\frac{1}{2}
    eg2: lim⁡x→0ln⁡sin⁡xx2=lim⁡x→0sin⁡xx−1x2=lim⁡x→0sin⁡x−xx3=lim⁡x→0−16x3x3=−16\lim\limits_{x\rightarrow 0}\frac{\ln\frac{\sin}{x}}{x^2} = \lim\limits_{x\rightarrow 0}\frac{\frac{\sin x}{x}-1}{x^2} = \lim\limits_{x\rightarrow 0}\frac{\sin x-x}{x^3} = \lim\limits_{x\rightarrow 0}\frac{-\frac{1}{6}x^3}{x^3}=-\frac{1}{6}

  3. (1+x)α−1∼αx(1+x)^\alpha-1 \sim \alpha x ,(α\alpha常以分数的形式出现)

    1. (1+f(x))α−1∼αf(x)(1+f(x))^\alpha-1 \sim \alpha f(x)

  4. 1−cos⁡x∼12x21-\cos x \sim \frac{1}{2}x^2

    1. 1−cos⁡f(x)∼12f2(x)1-\cos f(x) \sim \frac{1}{2}f^2(x)