求 $e^{\frac{1}{x + 2}}$ 的 Maclaurin 级数，展开到并包括 $x^2$ 项。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

先求函数在 $0$ 处的值和前两阶导数。二阶导数需要对一阶导数使用乘积法则。

For $f(x)=e^{1/(x+2)}$,

Here the product rule gives

Thus $f(0)=e^{1/2}$, $f'(0)=-\frac14e^{1/2}$ and $f''(0)=\frac5{16}e^{1/2}$. Using $f(x)=f(0)+xf'(0)+\frac{x^2}{2}f''(0)+O(x^3)$, the Maclaurin expansion is

也可先展开指数里的 $\frac1{x+2}$，提出常数对应的 $e^{1/2}$，再对余下的指数作展开。

Near $x=0$,

Writing $u=-\frac x4+\frac{x^2}{8}+O(x^3)$, we have $e^u=1+u+\frac12u^2+O(x^3)$. Therefore