常用泰勒级数

\(\begin{aligned} \frac{1}{1-x} &=1+x+x^{2}+x^{3}+x^{4}+\cdots \\ &=1+\sum_{k=1}^{\infty} x^{k} \end{aligned}\)

\(\begin{aligned}(1+x)^{\alpha}=1+\sum_{k=1}^{\infty}\left(\begin{array}{l}\alpha \\ k\end{array}\right) x^{k},\ \left(\begin{array}{l}\alpha \\ k\end{array}\right)=\frac{\alpha(\alpha-1) \cdots(\alpha-k+1)}{k !}\end{aligned}\)

\(\begin{aligned} e^{x} &=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\cdots \\ &=1+\sum_{k=1}^{\infty} \frac{x^{k}}{k !} \end{aligned}\)

\(\begin{aligned} \ln (1+x) &=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac{x^{5}}{5}-\cdots \\ &=\sum_{k=1}^{\infty}(-1)^{k-1} \frac{x^{k}}{k} \end{aligned}\)

\(\begin{aligned} \cos x &=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}-\cdots \\ &=1+\sum_{k=1}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !} \end{aligned}\)

\(\begin{aligned} \sin x &=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\cdots \\ &=\sum_{k=1}^{\infty}(-1)^{(k-1)} \frac{x^{2k-1}}{(2k-1) !} \end{aligned}\)

\(\begin{aligned} \tan ^{-1} x &=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\frac{x^{9}}{9}-\cdots \\ &=\sum_{k=1}^{\infty}(-1)^{k-1} \frac{x^{2k-1}}{2k-1} \end{aligned}\)