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常见函数的泰勒级数展开
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- 指数函数 e x e^x ex
e x = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ = ∑ n = 0 ∞ x n n ! e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!} ex=1+x+2!x2+3!x3+4!x4+⋯=n=0∑∞n!xn - 正弦函数 sin ( x ) \sin(x) sin(x)
sin ( x ) = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! \sin(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} sin(x)=x−3!x3+5!x5−7!x7+⋯=n=0∑∞(−1)n(2n+1)!x2n+1 - 余弦函数 cos ( x ) \cos(x) cos(x)
cos ( x ) = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! \cos(x) = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} cos(x)=1−2!x2+4!x4−6!x6+⋯=n=0∑∞(−1)n(2n)!x2n - 自然对数函数 ln ( 1 + x ) \ln(1+x) ln(1+x)
ln ( 1 + x ) = x − x 2 2 + x 3 3 − x 4 4 + ⋯ = ∑ n = 1 ∞ ( − 1 ) n + 1 x n n ( ∣ x ∣ < 1 ) \ln(1+x) = x – \frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \cdots = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \quad (|x| < 1) ln(1+x)=x−2x2+3x3−4x4+⋯=n=1∑∞(−1)n+1nxn(∣x∣<1) - 反正切函数 arctan ( x ) \arctan(x) arctan(x)
arctan ( x ) = x − x 3 3 + x 5 5 − x 7 7 + ⋯ = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 ( ∣ x ∣ ≤ 1 ) \arctan(x) = x – \frac{x^3}{3} + \frac{x^5}{5} – \frac{x^7}{7} + \cdots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \quad (|x| \leq 1) arctan(x)=x−3x3+5x5−7x7+⋯=n=0∑∞(−1)n2n+1x2n+1(∣x∣≤1) - 双曲正弦函数 sinh ( x ) \sinh(x) sinh(x)
sinh ( x ) = x + x 3 3 ! + x 5 5 ! + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! \sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} sinh(x)=x+3!x3+5!x5+⋯=n=0∑∞(2n+1)!x2n+1 - 双曲余弦函数 cosh ( x ) \cosh(x) cosh(x)
cosh ( x ) = 1 + x 2 2 ! + x 4 4 ! + ⋯ = ∑ n = 0 ∞ x 2 n ( 2 n ) ! \cosh(x) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} cosh(x)=1+2!x2+4!x4+⋯=n=0∑∞(2n)!x2n - ( 1 + x ) k (1 + x)^k (1+x)k 的泰勒级数（广义二项式定理）
- ( 1 + x ) k = 1 + k x + k ( k − 1 ) 2 ! x 2 + k ( k − 1 ) ( k − 2 ) 3 ! x 3 + ⋯ = ∑ n = 0 ∞ ( k n ) x n ( ∣ x ∣ < 1 ) (1 + x)^k = 1 + kx + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \cdots = \sum_{n=0}^{\infty} \binom{k}{n} x^n \quad (|x| < 1) (1+x)k=1+kx+2!k(k−1)x2+3!k(k−1)(k−2)x3+⋯=n=0∑∞(nk)xn(∣x∣<1)
- 对数函数 ln ( x ) \ln(x) ln(x)在 x = 1 x = 1 x=1 处展开：
ln ( x ) = ( x − 1 ) − ( x − 1 ) 2 2 + ( x − 1 ) 3 3 − ( x − 1 ) 4 4 + ⋯ = ∑ n = 1 ∞ ( − 1 ) n + 1 ( x − 1 ) n n ( 0 < x ≤ 2 ) \ln(x) = (x – 1) – \frac{(x – 1)^2}{2} + \frac{(x – 1)^3}{3} – \frac{(x – 1)^4}{4} + \cdots = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x – 1)^n}{n} \quad (0 < x \leq 2) ln(x)=(x−1)−2(x−1)2+3(x−1)3−4(x−1)4+⋯=n=1∑∞(−1)n+1n(x−1)n(0<x≤2) - 1 1 − x \frac{1}{1-x} 1−x1（几何级数）
1 1 − x = 1 + x + x 2 + x 3 + x 4 + ⋯ = ∑ n = 0 ∞ x n ( ∣ x ∣ < 1 ) \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots = \sum_{n=0}^{\infty} x^n \quad (|x| < 1) 1−x1=1+x+x2+x3+x4+⋯=n=0∑∞xn(∣x∣<1) - 反双曲正切函数 artanh ( x ) \text{artanh}(x) artanh(x) 或 tanh − 1 ( x ) \text{tanh}^{-1}(x) tanh−1(x)
artanh ( x ) = x + x 3 3 + x 5 5 + ⋯ = ∑ n = 0 ∞ x 2 n + 1 2 n + 1 ( ∣ x ∣ < 1 ) \text{artanh}(x) = x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} \quad (|x| < 1) artanh(x)=x+3x3+5x5+⋯=n=0∑∞2n+1x2n+1(∣x∣<1) - 反双曲余弦函数 arcosh ( x ) \text{arcosh}(x) arcosh(x)
arcosh ( x ) = ln ( x + x 2 − 1 ) \text{arcosh}(x) = \ln(x + \sqrt{x^2 – 1}) arcosh(x)=ln(x+x2−1)
展开式在 x = 1 x = 1 x=1 处较复杂，一般不直接使用泰勒展开。 - 反双曲正弦函数 arsinh ( x ) \text{arsinh}(x) arsinh(x)
arsinh ( x ) = x − x 3 6 + 3 x 5 40 − 5 x 7 112 + ⋯ \text{arsinh}(x) = x – \frac{x^3}{6} + \frac{3x^5}{40} – \frac{5x^7}{112} + \cdots arsinh(x)=x−6x3+403x5−1125x7+⋯ - 1 + x \sqrt{1+x} 1+x
1 + x = 1 + x 2 − x 2 8 + x 3 16 − ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( − 1 / 2 ) ( − 3 / 2 ) ( − 5 / 2 ) ⋯ ( − ( 2 n − 1 ) / 2 ) n ! x n ( ∣ x ∣ < 1 ) \sqrt{1+x} = 1 + \frac{x}{2} – \frac{x^2}{8} + \frac{x^3}{16} – \cdots = \sum_{n=0}^{\infty} (-1)^{n} \frac{(-1/2)(-3/2)(-5/2)\cdots(-(2n-1)/2)}{n!} x^n \quad (|x| < 1) 1+x=1+2x−8x2+16x3−⋯=n=0∑∞(−1)nn!(−1/2)(−3/2)(−5/2)⋯(−(2n−1)/2)xn(∣x∣<1) - 1 x \frac{1}{x} x1在 x = 1 x = 1 x=1 处展开：
1 x = 1 − ( x − 1 ) + ( x − 1 ) 2 − ( x − 1 ) 3 + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( x − 1 ) n ( 0 < x ≤ 2 ) \frac{1}{x} = 1 – (x – 1) + (x – 1)^2 – (x – 1)^3 + \cdots = \sum_{n=0}^{\infty} (-1)^n (x-1)^n \quad (0 < x \leq 2) x1=1−(x−1)+(x−1)2−(x−1)3+⋯=n=0∑∞(−1)n(x−1)n(0<x≤2) - arcsin ( x ) \arcsin(x) arcsin(x)
arcsin ( x ) = x + x 3 6 + 3 x 5 40 + ⋯ = ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 ( ∣ x ∣ ≤ 1 ) \arcsin(x) = x + \frac{x^3}{6} + \frac{3x^5}{40} + \cdots = \sum_{n=0}^{\infty} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} \quad (|x| \leq 1) arcsin(x)=x+6x3+403x5+⋯=n=0∑∞4n(n!)2(2n+1)(2n)!x2n+1(∣x∣≤1) - arccos ( x ) \arccos(x) arccos(x)
arccos ( x ) = π 2 − arcsin ( x ) = π 2 − ( x + x 3 6 + 3 x 5 40 + ⋯ ) \arccos(x) = \frac{\pi}{2} – \arcsin(x) = \frac{\pi}{2} – \left( x + \frac{x^3}{6} + \frac{3x^5}{40} + \cdots \right) arccos(x)=2π−arcsin(x)=2π−(x+6x3+403x5+⋯) - sec ( x ) \sec(x) sec(x)
sec ( x ) = 1 + x 2 2 + 5 x 4 24 + 61 x 6 720 + ⋯ \sec(x) = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + \cdots sec(x)=1+2x2+245x4+72061x6+⋯ - csc ( x ) \csc(x) csc(x)
csc ( x ) = 1 x + x 6 + 7 x 3 360 + ⋯ ( for 0 < ∣ x ∣ < π ) \csc(x) = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \cdots \quad (\text{for } 0 < |x| < \pi) csc(x)=x1+6x+3607x3+⋯(for 0<∣x∣<π)
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