泰勒 1/((1+x)^{1/2)}

解答

泰勒

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

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

求解步骤

应用泰勒公式

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

求导数

= 1 + \frac{- \frac{1}{2}}{1 !} x + \frac{\frac{3}{4}}{2 !} x^{2} + \frac{- \frac{15}{8}}{3 !} x^{3} + \frac{\frac{105}{16}}{4 !} x^{4} + \dots

整理后得

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

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