테일러 급수

$$\title{About Taylor Series}$$ $$ \put \begin{cases} f(x)&=\text{Any Function}\\ g(x)&=c_0+c_1x+c_2x^2+\cdots+c_nx^n+\cdots \end{cases} $$ $$ \put \begin{cases} f(0)&=g(0)\\ f'(0)&=g'(0)\\ f''(0)&=g''(0)\\ &\vdots\\ f^{(n)}(0)&=g^{(n)}(0)\\ &\vdots\\ \end{cases} $$ $$ \begin{cases} g(x)&=c_0+c_1x+c_2x^2+\cdots\\ g'(x)&=c_1+2c_2x+3c_3x^2+\cdots\\ g''(x)&=2c_2+3\cdot2c_3x+4\cdot3c_3x^2+\cdots\\ &\vdots\\ \end{cases} $$ $$ \begin{cases} g(0)&=c_0\\ g'(0)&=c_1\\ g''(0)&=2c_2\\ g'''(0)&=3\cdot2c_3\\ &\vdots\\ g^{(n)}(0)&=n!c_n\\ &\vdots\\ \end{cases} $$ $$ \begin{cases} c_0&=f(0)\\ c_1&=f'(0)\\ c_2&=\frac{f''(0)}{2!}\\ &\vdots\\ c_n&=\frac{f^{(n)}(0)}{n!}\\ &\vdots\\ \end{cases} $$ $$g(x)=c_0+c_1x+c_2x^2+\cdots+c_nx^n+\cdots$$ $$g(x)=f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\cdots+\frac{f^{(n)}(0)}{n!}x^n+\cdots$$ $$\therefore f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$$ $$\title{Exponential Function to Taylor Series}$$ $$\put \exp(x)=\e^x,$$ $$ \begin{aligned} \put f(x)&= \exp(x),\\ f'(x)&=f(x),\\ \end{aligned} $$ $$ \begin{aligned} \exp(x)&=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n\\ &=\sum_{n=0}^{\infty}\frac{\exp(0)}{n!}x^n\\ &=\sum_{n=0}^{\infty}\frac{x^n}{n!}\\ \end{aligned} $$