泰勒公式及余项

拉格朗日余项

f(x)=f(x0)+f′(x0)(x−x0)+⋯+1n!f(n)(x0)(x−x0)n+f(n+1)(ξ)(n+1)!(x−x0)n+1,ξ∈(x,x0)f(x) = f(x_0) + f'(x_0)(x-x_0) + \cdots + \frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}, \xi \in (x, x_0)

佩亚诺余项(x 无穷趋近于 x_0)

f(x)=f(x0)+f′(x0)(x−x0)+12!f′′(x0)(x−x0)+⋯+1n!f(n)(x0)(x−x0)n+o(x−x0)nf(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{1}{2!}f''(x_0)(x-x_0) + \cdots + \frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n + o(x-x_0)^n

麦克劳林公式(x0 = 0)

f(x)=f(0)+f′(0)x+f′′(0)2!x2+⋯+f(n)(0)n!xn+f(n+1)(ξ)(n+1)!xn+1f(x)=f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n+\frac{f^{(n+1)}(\xi)}{(n+1)!}x^{n+1} f(x)=f(0)+f′(0)x+f′′(0)2!x2+⋯+f(n)(0)n!xn+o(xn)f(x)=f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + o(x^n)