麦克劳林 1/((x^2-a^2))

解答

麦克劳林

\frac{1}{( x ^{2} - a ^{2} )}

- \frac{1}{a ^{2}} - \frac{1}{a ^{4}} x^{2} - \frac{1}{a ^{6}} x^{4} - \frac{1}{a ^{8}} x^{6} - \frac{1}{a ^{10}} x^{8} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= - \frac{1}{a ^{2}} + \frac{0}{1 !} x + \frac{- \frac{2}{a ^{4}}}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{- \frac{24}{a ^{6}}}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- \frac{720}{a ^{8}}}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{- \frac{40320}{a ^{10}}}{8 !} x^{8} + \dots

整理后得

= - \frac{1}{a ^{2}} - \frac{1}{a ^{4}} x^{2} - \frac{1}{a ^{6}} x^{4} - \frac{1}{a ^{8}} x^{6} - \frac{1}{a ^{10}} x^{8} + \dots

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