泰勒 1/(x-1),x=20

解答

泰勒

\frac{1}{x - 1}, x = 20

\frac{1}{19} - \frac{1}{361} (x - 20) + \frac{1}{6859} (x - 20)^{2} - \frac{1}{130321} (x - 20)^{3} + \frac{1}{2476099} (x - 20)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{19} + \frac{\frac{d}{d x} ( \frac{1}{x - 1} ) ( 20 )}{1 !} (x - 20) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{x - 1} ) ( 20 )}{2 !} (x - 20)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{x - 1} ) ( 20 )}{3 !} (x - 20)^{3} + \dots

求导数

= \frac{1}{19} + \frac{- \frac{1}{361}}{1 !} (x - 20) + \frac{\frac{2}{6859}}{2 !} (x - 20)^{2} + \frac{- \frac{6}{130321}}{3 !} (x - 20)^{3} + \frac{\frac{24}{2476099}}{4 !} (x - 20)^{4} + \dots

整理后得

= \frac{1}{19} - \frac{1}{361} (x - 20) + \frac{1}{6859} (x - 20)^{2} - \frac{1}{130321} (x - 20)^{3} + \frac{1}{2476099} (x - 20)^{4} + \dots

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