泰勒 x/(1-x^3)

解答

泰勒

\frac{x}{1 - x ^{3}}

x + x^{4} + x^{7} + x^{10} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{24}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{5040}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{0}{9 !} x^{9} + \frac{3628800}{10 !} x^{10} + \dots

整理后得

= x + x^{4} + x^{7} + x^{10} + \dots

\int \frac{4 e ^{3 x}}{1 + e ^{2 x}} d x

d er i v a t i v e \frac{( 8 x ^{3} - 4 x ^{2} + 3 )}{x}

x \to - 1 lim (3 x + 1)

s im pl i f y - 2 sin (t) cos (t)

\frac{d}{d x} ((x^{2} + y^{2} + z^{2})^{- \frac{1}{2}})