e x=∑n=0 ∞ 1n! x n= 1 + x +12! x 2+ ⋯ ∈ ( − ∞ , + ∞ ) sin ⁡ x=∑n=0 ∞(−1)n(2n+1)! x2n+1= x −13! x 3+15! x 5+ ⋯  , x ∈ ( − ∞ , + ∞ ) cos ⁡ x=∑n=0 ∞(−1)n(2n)! x2n= 1 −12! x 2+14! x 4+ ⋯  , x ∈ ( − ∞ , + ∞ ) ln ⁡ ( 1 + x )=∑n=0 ∞(−1)nn+1 xn+1= x −1 2 x 2+1 3 x 3+ ⋯  , x ∈ ( − 1 , 1 ] 11 − x =∑n=0 ∞ x n= 1 + x +x 2+x 3+ ⋯  , x ∈ ( − 1 , 1 ) 11 + x =∑n=0 ∞( − 1) n x n= 1 − x +x 2−x 3+ ⋯  , x ∈ ( − 1 , 1 ) ( 1 + x) α = 1 +∑n=1 ∞α(α−1)⋯(α−n+1)n! x n= 1 + α x + α(α−1)2! x 2+ ⋯  , x ∈ ( − 1 , 1 ) arctan ⁡ x=∑n=0 ∞(−1)n2n+1 x2n+1= x −1 3 x 3+1 5 x 5+ ⋯ + x ∈ [ − 1 , 1 ] arcsin ⁡ x=∑n=0 ∞(2n)!4n(n!)2(2n+1) x2n+1= x +1 6 x 3+3 40 x 5+5 112 x 7+35 1152 x 9+ ⋯ + , x ∈ ( − 1 , 1 ) tan ⁡ x=∑n=1 ∞B2 n (−4)n(1−4 n )(2n)! x2n−1= x +1 3 x 3+2 15 x 5+17 315 x 7+62 2835 x 9+1382 155925 x 11+21844 6081075 x 13+929569 638512875 x 15+ ⋯  , x ∈ ( −π 2,π 2) \begin{aligned} e^{x}&=\sum_{n=0}^{\infty} \frac{1}{n !} x^{n}=1+x+\frac{1}{2 !} x^{2}+\cdots \in(-\infty,+\infty) \\ \sin x&=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) !} x^{2 n+1}=x-\frac{1}{3 !} x^{3}+\frac{1}{5 !} x^{5}+\cdots, x \in(-\infty,+\infty) \\ \cos x&=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n) !} x^{2 n}=1-\frac{1}{2 !} x^{2}+\frac{1}{4 !} x^{4}+\cdots, x \in(-\infty,+\infty) \\ \ln (1+x)&=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{n+1}=x-\frac{1}{2} x^{2}+\frac{1}{3} x^{3}+\cdots, x \in(-1,1] \\ \frac{1}{1-x}&=\sum_{n=0}^{\infty} x^{n}=1+x+x^{2}+x^{3}+\cdots, x \in(-1,1) \\ \frac{1}{1+x}&=\sum_{n=0}^{\infty}(-1)^{n} x^{n}=1-x+x^{2}-x^{3}+\cdots, x \in(-1,1)\\ (1+x)^{\alpha}&=1+\sum_{n=1}^{\infty} \frac{\alpha(\alpha-1) \cdots(\alpha-n+1)}{n !} x^{n}=1+\alpha x+\frac{\alpha(\alpha-1)}{2 !} x^{2}+\cdots, x \in(-1,1) \\ \arctan x&=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2 n+1} x^{2 n+1}=x-\frac{1}{3} x^{3}+\frac{1}{5} x^{5}+\cdots+ x \in[-1,1] \\ \arcsin x&=\sum_{n=0}^{\infty} \frac{(2 n) !}{4^{n}(n !)^{2}(2 n+1)} x^{2n+1}=x+\frac{1}{6} x^{3}+\frac{3}{40} x^{5}+\frac{5}{112} x^{7}+\frac{35}{1152} x^{9}+\cdots+, x \in(-1,1)\\ \tan x&=\sum_{n=1}^{\infty} \frac{B_{2 n}(-4)^{n}\left(1-4^{n}\right)}{(2 n) !} x^{2 n-1}=x+\frac{1}{3} x^{3}+\frac{2}{15} x^{5}+\frac{17}{315} x^{7}+\frac{62}{2835} x^{9}+\frac{1382}{155925} x^{11}+\frac{21844}{6081075} x^{13}+\frac{929569}{638512875} x^{15}+\cdots,x\in (-\frac{\pi}{2},\frac{\pi}{2}) \end{aligned}exsinxcosxln(1+x)1−x1​1+x1​(1+x)αarctanxarcsinxtanx​=n=0∑∞​n!1​xn=1+x+2!1​x2+⋯∈(−∞,+∞)=n=0∑∞​(2n+1)!(−1)n​x2n+1=x−3!1​x3+5!1​x5+⋯,x∈(−∞,+∞)=n=0∑∞​(2n)!(−1)n​x2n=1−2!1​x2+4!1​x4+⋯,x∈(−∞,+∞)=n=0∑∞​n+1(−1)n​xn+1=x−21​x2+31​x3+⋯,x∈(−1,1]=n=0∑∞​xn=1+x+x2+x3+⋯,x∈(−1,1)=n=0∑∞​(−1)nxn=1−x+x2−x3+⋯,x∈(−1,1)=1+n=1∑∞​n!α(α−1)⋯(α−n+1)​xn=1+αx+2!α(α−1)​x2+⋯,x∈(−1,1)=n=0∑∞​2n+1(−1)n​x2n+1=x−31​x3+51​x5+⋯+x∈[−1,1]=n=0∑∞​4n(n!)2(2n+1)(2n)!​x2n+1=x+61​x3+403​x5+1125​x7+115235​x9+⋯+,x∈(−1,1)=n=1∑∞​(2n)!B2n​(−4)n(1−4n)​x2n−1=x+31​x3+152​x5+31517​x7+283562​x9+1559251382​x11+608107521844​x13+638512875929569​x15+⋯,x∈(−2π​,2π​)​

Markdown 代码如下：

相关