詳解：馬克勞林級數展開式為

\(f(x) \buildrel\textstyle.\over= f(0) + \frac{{f'(0)}}{{1!}}x + \frac{{f''(0)}}{{2!}}{x^2} + \frac{{f'''(0)}}{{3!}}{x^3} +  \cdots \)

\(f(x) = {e^x}\) \(  \Rightarrow f(0) = {e^0} = 1\)

\(f'(x) = {e^x}\) \(  \Rightarrow f'(0) = {e^0} = 1\)

\(f''(x) = {e^x}\) \(  \Rightarrow f''(0) = {e^0} = 1\)

\(f'''(x) = {e^x}\) \(  \Rightarrow f'''(0) = {e^0} = 1\)

\({e^x} \buildrel\textstyle.\over= 1 + \frac{1}{{1!}}x + \frac{1}{{2!}}{x^2} + \frac{1}{{3!}}{x^3} +  \cdots \)