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泰勒 ln(4-x)

解答

泰勒

ln (4 - x)

解答

2 ln (2) - \frac{1}{4} x - \frac{1}{32} x^{2} - \frac{1}{192} x^{3} - \frac{1}{1024} x^{4} + \dots

求解步骤

应用泰勒公式

= 2 ln (2) + \frac{\frac{d}{d x} ( ln ( 4 - x ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ln ( 4 - x ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ln ( 4 - x ) ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 2 ln (2) + \frac{- \frac{1}{4}}{1 !} x + \frac{- \frac{1}{16}}{2 !} x^{2} + \frac{- \frac{1}{32}}{3 !} x^{3} + \frac{- \frac{3}{128}}{4 !} x^{4} + \dots

整理后得

= 2 ln (2) - \frac{1}{4} x - \frac{1}{32} x^{2} - \frac{1}{192} x^{3} - \frac{1}{1024} x^{4} + \dots

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