更多展开式可以查看Wolfram。

$$
\begin{aligned}
e^x &=& 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + o(x^3) \\
\alpha^x &=& 1 + \ln \alpha x + \frac{\ln^2 \alpha}{2!}x^2 + \frac{\ln^3 \alpha}{3!}x^3 + o(x^3)\\
\ln (x+1) &=& x – \frac{1}{2}x^2 + \frac{1}{3} x^3 + o(x^3) \\
\sqrt{1+x} &=& 1 + \frac{1}{2}x – \frac{1}{8}x^2 + \frac{1}{16}x^3 + o(x^3)\\
(1+x)^{\alpha} &=& 1 + \alpha x + \frac{\alpha(\alpha – 1)}{2!}x^2 + \frac{\alpha(\alpha – 1)(\alpha – 2)}{3!}x^3 + o(x^3)\\
\frac{1}{1+x} &=& 1-x+x^2-x^3+o(x^3)\\
\sin x &=& x – \frac{1}{3!}x^3 + \frac{1}{5!}x^5 + o(x^5)\\
\arcsin x &=& x + \frac{1}{2\times 3}x^3 + \frac{1\times 3}{2\times 4\times 5}x^5 + \frac{1\times 3\times 5}{2\times 4\times 6\times 7}x^7 + o(x^7)\\
\cos x &=& 1 – \frac{1}{2!}x^2 + \frac{1}{4!}x^4 + o(x^4)\\
\arccos x &=& \frac{\pi}{2} – x – \frac{1}{6} x^3 + o(x^3) \quad (\arcsin x + \arccos x = \frac{\pi}{2})\\
\tan x &=& x + \frac{1}{3}x^3 + \frac{2}{15}x^5 + o(x^5)\\
\arctan x &=& x – \frac{1}{3}x^3 + \frac{1}{5}x^5 + o(x^5) \\
\end{aligned}
$$