泰勒 4/x ,3

解答

泰勒

\frac{4}{x}, 3

\frac{4}{3} - \frac{4}{9} (x - 3) + \frac{4}{27} (x - 3)^{2} - \frac{4}{81} (x - 3)^{3} + \frac{4}{243} (x - 3)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= \frac{4}{3} + \frac{- \frac{4}{9}}{1 !} (x - 3) + \frac{\frac{8}{27}}{2 !} (x - 3)^{2} + \frac{- \frac{8}{27}}{3 !} (x - 3)^{3} + \frac{\frac{32}{81}}{4 !} (x - 3)^{4} + \dots

整理后得

= \frac{4}{3} - \frac{4}{9} (x - 3) + \frac{4}{27} (x - 3)^{2} - \frac{4}{81} (x - 3)^{3} + \frac{4}{243} (x - 3)^{4} + \dots

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