数二 随笔

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$$\begin{array}{}\text{(1)}& sinx=x-\frac{x^3}{3!}+o(x^3)\end{array}$$$$\begin{array}{}\text{(2)}& cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}+o(x^4)\end{array}$$$$\begin{array}{}\text{(3)}& arcsinx=x+\frac{x^3}{3!}+o(x^3)\end{array}$$$$\begin{array}{}\text{(4)}& tanx=x+\frac{x^3}{3}+o(x^3)\end{array}$$$$\begin{array}{}\text{(5)}& arctanx=x-\frac{x^3}{3}+o(x^3)\end{array}$$$$\begin{array}{}\text{(6)}& ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+o(x^3)\end{array}$$$$\begin{array}{}\text{(7)}& e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+o(x^3)\end{array}$$$$\begin{array}{}\text{(8)}& (1+x{)}^a=1+ax+\frac{a(a-1)}{2!}{x}^2+o(x^2)\end{array}$$