泰勒多项式
泰勒展开
f ( x ) = P n ( x ) + R n ( x ) f(x)=P_n(x)+R_n(x) f(x)=Pn(x)+Rn(x)
P n ( x ) = ∑ 0 n f ( k ) ( x 0 ) k ! ( x − x 0 ) k P_n(x)=\sum_0^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k Pn(x)=∑0nk!f(k)(x0)(x−x0)k
R n ( x ) = f ( n + 1 ) ( ξ x 0 ) ( n + 1 ) ! ( x − x 0 ) n + 1 R_n(x)=\frac{f^{(n+1)}(\xi{x_0})}{(n+1)!}(x-x_0)^{n+1} Rn(x)=(n+1)!f(n+1)(ξx0)(x−x0)n+1
这表示把函数展开成一个多项式的和以及一个误差项