\begin{array}{|c|c|}
\hline
\text{函数} & \text{泰勒展开式} \\
\hline
e^x & e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \\
\hline
\sin x & \sin x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{(2k+1)!} \\
\hline
\cos x & \cos x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{(2k)!} \\
\hline
\ln(1+x) & \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \\
\hline
\frac{1}{1-x} & \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \\
\hline
(1+x)^\alpha & (1+x)^\alpha = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^n, \text{其中}\binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!} \\
\hline
\tan x & \tan x = \sum_{k=1}^{\infty} \frac{(-1)^{k-1} 2^{2k} (2^{2k}-1) B_{2k}}{(2k)!} x^{2k-1}, \text{对于奇次项，}B_{2k}\text{是伯努利数} \\
\hline
\arctan x & \arctan x = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k+1}}{2k+1} \\
\hline
\sinh x & \sinh x = \sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!} \\
\hline
\cosh x & \cosh x = \sum_{k=0}^{\infty} \frac{x^{2k}}{(2k)!} \\
\hline
\end{array}