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泰勒 ln(3-x)

解答

泰勒

ln (3 - x)

解答

ln (3) - \frac{1}{3} x - \frac{1}{18} x^{2} - \frac{1}{81} x^{3} - \frac{1}{324} x^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= ln (3) + \frac{- \frac{1}{3}}{1 !} x + \frac{- \frac{1}{9}}{2 !} x^{2} + \frac{- \frac{2}{27}}{3 !} x^{3} + \frac{- \frac{2}{27}}{4 !} x^{4} + \dots

整理后得

= ln (3) - \frac{1}{3} x - \frac{1}{18} x^{2} - \frac{1}{81} x^{3} - \frac{1}{324} x^{4} + \dots

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