テイラー展開

Taylor expansion

$ f(x)=f(x_0)+\frac{d}{dx}f(x_0)(x-x_0)+\frac{1}{2!}\frac{d^2}{dx^2}f(x_0)(x-x_0)^2+\frac{1}{3!}\frac{d^3}{dx^3}f(x_0)(x-x_0)^3+O((x-x_0)^4)

$ r=\lim_{n\to\infty}\large \left|\frac{\frac{f^{(n)}(x_0)}{n!}}{\frac{f^{(n+1)}(x_0)}{(n+1)!}}\right|