Peano 余项泰勒公式
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\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
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\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)
常用 Peano 余项泰勒公式
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\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)
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