麦克劳林 f(x)=e^{x/2}

解答

麦克劳林

f (x) = e^{\frac{x}{2}}

1 + \frac{1}{2} x + \frac{1}{8} x^{2} + \frac{1}{48} x^{3} + \frac{1}{384} x^{4} + \dots

求解步骤

应用麦克劳林公式

= 1 + \frac{\frac{d}{d x} ( e ^{\frac{x}{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{\frac{x}{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{\frac{x}{2}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 1 + \frac{\frac{1}{2}}{1 !} x + \frac{\frac{1}{4}}{2 !} x^{2} + \frac{\frac{1}{8}}{3 !} x^{3} + \frac{\frac{1}{16}}{4 !} x^{4} + \dots

整理后得

= 1 + \frac{1}{2} x + \frac{1}{8} x^{2} + \frac{1}{48} x^{3} + \frac{1}{384} x^{4} + \dots

in v erse l a pl a ce {\frac{( s + 1 ) ^{2}}{s ^{3}}}

y^{^{'}} = \frac{x y ^{3}}{25}

\frac{\partial}{\partial y} (1 - y^{2})

d er i v a t i v e 2 sin (x)

\int \frac{2 x + 1}{e ^{2 x}} d x