常用 Peano 余项泰勒公式

f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + f ′ ′ ( x 0 ) 2 ! ( x − x 0 ) 2 + ⋯ + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + o ( ( x − x 0 ) n ) \large f(x)=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+\frac{f^{\prime \prime}\left(x_{0}\right)}{2 !}\left(x-x_{0}\right)^{2}+\cdots+\frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}+o\left(\left(x-x_{0}\right)^{n}\right) f(x)=f(x0​)+f′(x0​)(x−x0​)+2!f′′(x0​)​(x−x0​)2+⋯+n!f(n)(x0​)​(x−x0​)n+o((x−x0​)n) 特别是当 x 0 = 0 x_{0}=0 x0​=0 时， f ( x ) = f ( 0 ) + f ′ ( 0 ) x + f ′ ′ ( 0 ) 2 ! x 2 + ⋯ + f ( n ) ( 0 ) n ! x n + o ( x n ) \huge f(x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^{2}+\cdots+\frac{f^{(n)}(0)}{n !} x^{n}+o\left(x^{n}\right) f(x)=f(0)+f′(0)x+2!f′′(0)​x2+⋯+n!f(n)(0)​xn+o(xn)

e x = 1 + x + x 2 ! + ⋯ + x n ! + o ( x n ) sin ⁡ x = x − x 3 3 ! + ⋯ + ( − 1 ) n − 1 x 2 n − 1 ( 2 n − 1 ) ! + o ( x 2 n − 1 ) cos ⁡ x = 1 − x 2 2 ! + ⋯ + ( − 1 ) n x 2 n ( 2 n ) ! + o ( x 2 n ) ln ⁡ ( 1 + x ) = x − x 2 2 + ⋯ + ( − 1 ) n − 1 x n n + o ( x n ) . ( 1 + x ) α = 1 + α x + α ( α − 1 ) 21 x 2 + ⋯ + α ( α − 1 ) ⋯ ( α − n + 1 ) n ! x n + o ( x n ) \Large \begin{array}{l} \mathrm{e}^{x}=1+x+\frac{x}{2 !}+\cdots+\frac{x}{n !}+o\left(x^{n}\right) \\ \sin x=x-\frac{x^{3}}{3 !}+\cdots+(-1)^{n-1} \frac{x^{2 n-1}}{(2 n-1) !}+o\left(x^{2 n-1}\right) \\ \cos x=1-\frac{x^{2}}{2 !}+\cdots+(-1)^{n} \frac{x^{2 n}}{(2 n) !}+o\left(x^{2 n}\right) \\ \ln (1+x)=x-\frac{x^{2}}{2}+\cdots+(-1)^{n-1} \frac{x^{n}}{n}+o\left(x^{n}\right) . \\ (1+x)^{\alpha}=1+\alpha x+\frac{\alpha(\alpha-1)}{21} x^{2}+\cdots+\frac{\alpha(\alpha-1) \cdots(\alpha-n+1)}{n !} x^{n}+o\left(x^{n}\right) \end{array} ex=1+x+2!x​+⋯+n!x​+o(xn)sinx=x−3!x3​+⋯+(−1)n−1(2n−1)!x2n−1​+o(x2n−1)cosx=1−2!x2​+⋯+(−1)n(2n)!x2n​+o(x2n)ln(1+x)=x−2x2​+⋯+(−1)n−1nxn​+o(xn).(1+x)α=1+αx+21α(α−1)​x2+⋯+n!α(α−1)⋯(α−n+1)​xn+o(xn)​