麦克劳林 (1-x^2)^{1/2}

解答

麦克劳林

(1 - x^{2})^{\frac{1}{2}}

1 - \frac{1}{2} x^{2} - \frac{1}{8} x^{4} - \frac{1}{16} x^{6} - \frac{5}{128} x^{8} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= 1 + \frac{0}{1 !} x + \frac{- 1}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- 3}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 45}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{- 1575}{8 !} x^{8} + \dots

整理后得

= 1 - \frac{1}{2} x^{2} - \frac{1}{8} x^{4} - \frac{1}{16} x^{6} - \frac{5}{128} x^{8} + \dots

2 x y \frac{d y}{d x} = 3

\int \frac{ln ( x ) - 1}{x} d x

x \to 1 lim (3 x - 6)

\frac{d}{d x} (ln (6 \frac{x - 5}{x + 5}))

in v erse l a pl a ce \frac{2 s e ^{- 2 s}}{5 s ^{2} + 45}