泰勒 e^{e^t}

解答

泰勒

e^{e^{t}}

e + e t + e t^{2} + \frac{5 e}{6} t^{3} + \frac{5 e}{8} t^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= e + \frac{e}{1 !} t + \frac{2 e}{2 !} t^{2} + \frac{5 e}{3 !} t^{3} + \frac{15 e}{4 !} t^{4} + \dots

整理后得

= e + e t + e t^{2} + \frac{5 e}{6} t^{3} + \frac{5 e}{8} t^{4} + \dots

\int 5 sec^{2} (5 x) tan^{5} (5 x) d x

\frac{d}{d x} (\frac{( x + 1 )}{2})

\frac{d}{d t} (a (1 - cos (t)))

\frac{d y}{d t} = e^{t} (y - 1)^{2}

x \to 2 lim (\frac{x + 2}{x ^{2} - 4})