麦克劳林 (1+x)^{-5/2}

解答

麦克劳林

(1 + x)^{- \frac{5}{2}}

1 - \frac{5}{2} x + \frac{35}{8} x^{2} - \frac{105}{16} x^{3} + \frac{1155}{128} x^{4} + \dots

求解步骤

应用麦克劳林公式

= 1 + \frac{\frac{d}{d x} ( ( 1 + x ) ^{- \frac{5}{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ( 1 + x ) ^{- \frac{5}{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( 1 + x ) ^{- \frac{5}{2}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 1 + \frac{- \frac{5}{2}}{1 !} x + \frac{\frac{35}{4}}{2 !} x^{2} + \frac{- \frac{315}{8}}{3 !} x^{3} + \frac{\frac{3465}{16}}{4 !} x^{4} + \dots

整理后得

= 1 - \frac{5}{2} x + \frac{35}{8} x^{2} - \frac{105}{16} x^{3} + \frac{1155}{128} x^{4} + \dots

x \to 6 lim (\frac{1}{( x - 6 ) ^{2}})

a re a y = x^{3}, y = 1, x = 2, x = 2

\int \frac{1}{3 x - 5} d x

x \to 0 - lim (\frac{3}{x} - \frac{3}{∣ x ∣})

a re a e^{x}, e^{- x}, - 2, 1