泰勒公式

1. 指数函数 ( $ e^x $ )

$$
e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots = \sum_{n=0}^\infty \frac{x^n}{n!}
$$

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2. 正弦函数 ($ \sin x$ )

$$
\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}
$$

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3. 余弦函数 ($ \cos x$ )

$$
\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}
$$

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4. 自然对数函数 ($ \ln(1 + x)$ )

$$
\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n}, \quad |x| < 1
$$

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5. 几何级数 ( $\frac{1}{1 - x} $)

$$
\frac{1}{1 - x} = 1 + x + x^2 + x^3 + x^4 + \cdots = \sum_{n=0}^\infty x^n, \quad |x| < 1
$$

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6. 反正切函数 ( $\arctan x $)

$$
\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}, \quad |x| \leq 1
$$

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7. 二项式级数 ($ (1 + x)^k $)（( k ) 为任意实数）

$$
(1 + x)^k = 1 + kx + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \cdots = \sum_{n=0}^\infty \binom{k}{n} x^n, \quad |x| < 1
$$

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8. 双曲正弦函数 ($ \sinh x $)

$$
\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}
$$

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9. 双曲余弦函数 ( $\cosh x $)

$$
\cosh x = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}
$$

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10. 平方根函数 ( $\sqrt{1 + x}$ )

$$
\sqrt{1 + x} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128}x^4 + \cdots, \quad |x| < 1
$$

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11. 正切函数

$$
\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{135}
$$