Taylor Series

Taylor Series around the point $x_0 = 0$ for basic elementary functions.

$$e^x=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+o(x^n)=\sum\limits_{k=0}^n\frac{x^k}{k!}+o(x^k);\quad x\rightarrow 0$$

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+...+\frac{(-1)^nx^{2n+1}}{(2n+1)!}+o(x^{2n+2})=\sum\limits_{k=0}^n(-1)^k\frac{x^{2k+1}}{(2k+1)!}+o(x^{2n+2});\quad x\rightarrow 0y$$

$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}+...+(-1)^n\frac{x^{2n}}{(2n)!}+o(x^{2n+1})=\sum\limits_{k=0}^n(-1)^k\frac{x^{2k}}{(2k)!}+o(x^{2n+1});\quad x\rightarrow 0$$

$$(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+...+\frac{\alpha(\alpha-1)...(\alpha-(n-1))}{n!}x^n+o(x^n);\quad x\rightarrow 0$$

$$\frac{1}{1+x}=\sum\limits_{k=0}^n (-1)^kx^k+o(x^n);\quad x\rightarrow 0$$

$$\frac{1}{1-x}=\sum\limits_{k=0}^n x^k+o(x^n);\quad x\rightarrow 0$$

$$\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+...+\frac{(-1)^{n-1}x^n}{n}+o(x^n)=\sum\limits_{k=0}^n \frac{(-1)^{k-1}x^k}{k}+o(x^n);\quad x\rightarrow 0$$

$$\log(1-x)=-\sum\limits_{k=0}^n \frac{x^k}{k}+o(x^n)\quad x\rightarrow 0.$$

$$tg x = x+\frac{x^3}{3}+\frac{x^5}{15}+\overline{o}(x^5) \quad x\rightarrow 0$$

$$\arcsin x = x+\sum\limits_{k=1}^n\frac{(2k-1)!!}{2^k k!(2k+1)}x^{2k+1}+\overline{o}(x^{2n+2}), \quad x\rightarrow 0$$