📝 题目
例 8 试将函数 $f\left( x\right) = {\mathrm{e}}^{\cos x}$ 在原点展开到 4 阶项.
💡 答案与解析
解 我们有
$$ {\mathrm{e}}^{\cos x} = \mathrm{e} \cdot {\mathrm{e}}^{\cos x - 1} $$
$$ = \mathrm{e}\left\lbrack {1 + \left( {\cos x - 1}\right) + \frac{1}{2}{\left( \cos x - 1\right) }^{2}}\right. $$
$$ \left. {+o\left( {\left( \cos x - 1\right) }^{2}\right) }\right\rbrack $$
$$ = \mathrm{e}\left\lbrack {1 + \left( {-\frac{{x}^{2}}{2} + \frac{{x}^{4}}{24} + o\left( {x}^{4}\right) }\right) }\right. $$
$$ \left. {+\frac{1}{2}{\left( -\frac{{x}^{2}}{2} + o\left( {x}^{2}\right) \right) }^{2} + o\left( {\left( -\frac{{x}^{2}}{2} + o\left( {x}^{2}\right) \right) }^{2}\right) }\right\rbrack $$
$$ = \mathrm{e}\left\lbrack {1 - \frac{{x}^{2}}{2} + \frac{{x}^{4}}{6} + o\left( {x}^{4}\right) }\right\rbrack $$
$$ = \mathrm{e} - \frac{\mathrm{e}}{2}{x}^{2} + \frac{\mathrm{e}}{6}{x}^{4} + o\left( {x}^{4}\right) . $$