泰勒 (-x+ln(1+x))/(x^2)

解答

泰勒

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

- 1 + ln (2) + \frac{- 4 ln ( 2 ) + 3}{2} (x - 1) + \frac{24 ln ( 2 ) - 17}{8} (x - 1)^{2} + \frac{- 192 ln ( 2 ) + 134}{48} (x - 1)^{3} + \frac{1920 ln ( 2 ) - 1334}{384} (x - 1)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= - 1 + ln (2) + \frac{\frac{- 4 l n ( 2 ) + 3}{2}}{1 !} (x - 1) + \frac{\frac{24 l n ( 2 ) - 17}{4}}{2 !} (x - 1)^{2} + \frac{\frac{- 192 l n ( 2 ) + 134}{8}}{3 !} (x - 1)^{3} + \frac{\frac{1920 l n ( 2 ) - 1334}{16}}{4 !} (x - 1)^{4} + \dots

整理后得

= - 1 + ln (2) + \frac{- 4 ln ( 2 ) + 3}{2} (x - 1) + \frac{24 ln ( 2 ) - 17}{8} (x - 1)^{2} + \frac{- 192 ln ( 2 ) + 134}{48} (x - 1)^{3} + \frac{1920 ln ( 2 ) - 1334}{384} (x - 1)^{4} + \dots

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