$$e^x=1+x+\frac{x^2}{2!}+…+\frac{x^n}{n!}+…=\sum_{n=0}^{\infty} \frac{x^n}{n!}$$
$$ sinx = x-\frac{1}{3!}x^3+…+(-1)^n\frac{1}{(2n+1)!}x^{2n+1}+…=\sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n+1}}{(2n+1)!} $$
$$ cosx = 1 - \frac{1}{2!}x^2 +…+ (-1)^n\frac{1}{(2n)!}x^{2n}+…=\sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n}}{(2n)!} $$
$$ ln(1+x)=x-\frac{1}{2}x^2+…+(-1)^{n-1}\frac{x^n}{n}+…=\sum_{n=0}^{\infty} (-1)^{n-1}\frac{x^{n}}{n}, -1 \lt x \le 1 $$
$$ (1+x)^{\alpha} = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2} x^2 + o(x^2) (x \rightarrow 0, a \ne 0) $$
$$ tanx = x + \frac{1}{3}x^3+o(x^3) (x \rightarrow 0) $$
$$ arcsinx = x + \frac{1}{6}x^3+o(x^3) (x \rightarrow 0) $$
$$ arctanx = x - \frac{1}{3}x^3+o(x^3) (x \rightarrow 0) $$