📝 题目
例 3 求函数 $f\left( x\right) = \ln \left( {1 + x}\right)$ 的麦克劳林公式.
💡 答案与解析
解 我们有
$$ f\left( x\right) = \ln \left( {1 + x}\right) , $$
$$ f\left( 0\right) = 0, $$
$$ {f}^{\prime }\left( x\right) = \frac{1}{1 + x}, $$
$$ {f}^{\prime }\left( 0\right) = 1, $$
$$ {f}^{\left( k\right) }\left( x\right) = {\left( -1\right) }^{k - 1}\frac{\left( {k - 1}\right) !}{{\left( 1 + x\right) }^{k}}, $$
$$ {f}^{\left( k\right) }\left( 0\right) = {\left( -1\right) }^{k - 1}\left( {k - 1}\right) ! $$
$$ \left( {k = 2,3,\cdots }\right) \text{ . } $$
于是得出
$$ \ln \left( {1 + x}\right) = x - \frac{{x}^{2}}{2} + \frac{{x}^{3}}{3} - \cdots + {\left( -1\right) }^{n - 1}\frac{{x}^{n}}{n} + o\left( {x}^{n}\right) . $$