Loading... 1. $\color{Red} e^x$ $$ e^x = \sum\limits_{n=0}^{\infty} \frac{1}{n!}x^n = 1+x+\frac{1}{2!}x^2+\ ...\ +\frac{1}{n!}x^n+\ ...,\ x\in(-\infty,+\infty) $$ 2. $\color{Red} \sin x$ $$ \sin x =\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\ ...\ +\frac{(-1)^n}{(2n+1)!}x^{2n+1}+...,x\in(-\infty,+\infty) $$ 3. $\color{red} \cos x$ $$ \cos x =\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\ ...\ +\frac{(-1)^n}{(2n)!}x^{2n}+...,x\in(-\infty,+\infty) $$ 4. $\color{red} \ln (1+x)$ $$ \ln (1+x) =\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{n+1}x^{n+1}=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\ ...\ +\frac{(-1)^n}{n+1}x^{n+1}+\ ...,x\in(-1,1\rbrack $$ 5. $\color{red} -\ln (1-x)$ $$ -\ln (1-x) =\sum\limits_{n=1}^{\infty}\frac{x^n}{n},x\in\lbrack -1,1) $$ 5. $\color{red} \frac{1}{1-x}$ $$ \frac{1}{1-x}=\sum\limits_{n=0}^{\infty}x^n = 1+x+x^2+x^3+\ ...\ +x^n+\ ...,x\in(-1,1) $$ 6. $\color{red} \frac{1}{1+x}$ $$ \frac{1}{1+x}=\sum\limits_{n=0}^{\infty}(-1)^nx^n=1-x+x^2-x^3+\ ...\ +(-1)^nx^n+\ ...,x\in(-1,1) $$ 7. $\color{red} (1+x)^\alpha$ $$ (1+x)^\alpha = 1+\sum\limits_{n=1}^{\infty}\frac{\alpha (\alpha -1)...(\alpha -n+1)}{n!}x^n = 1+\alpha x+\frac{\alpha (\alpha-1)}{2!}x^2+\ ...\ +\frac{\alpha (\alpha -1)...(\alpha -n+1)}{n!}x^n+\ ...,x\in(-1,1)|\alpha \ne 0 $$ 8. $\color{red} \arctan x$ $$ \arctan x =\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{2n+1}x^{2n+1}=x-\frac{1}{3}x^3+\frac{1}{5}x^5+\ ...\ +\frac{(-1)^n}{2n+1}x^{2n+1}+\ ...,x\in(-1,1) $$ 9. $\color{red} \arcsin x$ $$ \arcsin x =x+\frac{1}{6}x^3+\frac{3}{40}x^5+\ ...,x\in(-1,1) $$ 10. $\color{red} \tan x$ $$ \tan x =x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+\frac{62}{2835}x^9+\frac{1382}{155925}x^{11}+\ ...,x\in(-1,1) $$ 11. $\color{red} \sec x$ $$ \sec x =1+\frac{1}{2}x^2+\frac{5}{24}x^4+\frac{61}{720}x^6+\ ...,x\in(-\frac{\pi}{2},\frac{\pi}{2}) $$ 12. $\color{red} \frac{e^x-e^{-x}}{2}$ $$ \frac{e^x-e^{-x}}{2} = \sum\limits_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!},x\in(-\infty , +\infty) $$ 13. $\color{red} \frac{e^x+e^{-x}}{2}$ $$ \frac{e^x+e^{-x}}{2} = \sum\limits_{n=0}^{\infty}\frac{x^{2n}}{(2n)!},x\in(-\infty , +\infty) $$ 14. $\color{red} \frac{x}{(1-x)^2}$ $$ \frac{x}{(1-x)^2} = \sum\limits_{n=1}^{\infty}nx^n,x\in(-1,1) $$ 最后修改：2023 年 05 月 30 日 © 允许规范转载 赞 1 如果觉得我的文章对你有用，请随意赞赏
