泰勒 2/x ,3

解答

泰勒

\frac{2}{x}, 3

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

求解步骤

应用泰勒公式

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

求导数

= \frac{2}{3} + \frac{- \frac{2}{9}}{1 !} (x - 3) + \frac{\frac{4}{27}}{2 !} (x - 3)^{2} + \frac{- \frac{4}{27}}{3 !} (x - 3)^{3} + \frac{\frac{16}{81}}{4 !} (x - 3)^{4} + \dots

整理后得

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

x^{2} y^{^{''}} + x y^{^{'}} - y = \frac{1}{x}

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\frac{d}{d x} (lo g_{10} ((arcsin (x))^{2}))