泰勒 1/x

解答

泰勒

\frac{1}{x}

1 - (x - 1) + (x - 1)^{2} - (x - 1)^{3} + (x - 1)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= 1 + \frac{- 1}{1 !} (x - 1) + \frac{2}{2 !} (x - 1)^{2} + \frac{- 6}{3 !} (x - 1)^{3} + \frac{24}{4 !} (x - 1)^{4} + \dots

整理后得

= 1 - (x - 1) + (x - 1)^{2} - (x - 1)^{3} + (x - 1)^{4} + \dots

\int (4 x - \frac{1}{2 x}) d x

\int \frac{1}{y ^{2} - 16} d y

y^{^{'}} = t^{\frac{1}{4}} - y

\int_{- 7}^{7} (49 - x^{2}) d x

\frac{d}{d x} (ln (\frac{x + 3}{e ^{3 x}}))