泰勒 (2x)/(sqrt(e^{2x^2)-1)}

解答

泰勒

\frac{2 x}{e ^{2 x^{2}} - 1}

\frac{2}{e ^{2} - 1} + \frac{2 ( - e ^{2} - 1 )}{( e ^{2} - 1 ) e ^{2} - 1} (x - 1) - \frac{4 e ^{4} - 28 e ^{2}}{2 ( e ^{2} - 1 ) ^{\frac{5}{2}}} (x - 1)^{2} - \frac{- 20 e ^{6} + 88 e ^{4} + 172 e ^{2}}{6 ( e ^{2} - 1 ) ^{\frac{7}{2}}} (x - 1)^{3} - \frac{8 e ^{8} + 288 e ^{6} - 2520 e ^{4} - 1136 e ^{2}}{24 ( e ^{2} - 1 ) ^{\frac{9}{2}}} (x - 1)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= \frac{2}{e ^{2} - 1} + \frac{\frac{2 ( - e ^{2} - 1 )}{( e ^{2} - 1 ) e ^{2} - 1}}{1 !} (x - 1) + \frac{- \frac{4 e ^{4} - 28 e ^{2}}{( e ^{2} - 1 ) ^{\frac{5}{2}}}}{2 !} (x - 1)^{2} + \frac{- \frac{- 20 e ^{6} + 88 e ^{4} + 172 e ^{2}}{( e ^{2} - 1 ) ^{\frac{7}{2}}}}{3 !} (x - 1)^{3} + \frac{- \frac{8 e ^{8} + 288 e ^{6} - 2520 e ^{4} - 1136 e ^{2}}{( e ^{2} - 1 ) ^{\frac{9}{2}}}}{4 !} (x - 1)^{4} + \dots

整理后得

= \frac{2}{e ^{2} - 1} + \frac{2 ( - e ^{2} - 1 )}{( e ^{2} - 1 ) e ^{2} - 1} (x - 1) - \frac{4 e ^{4} - 28 e ^{2}}{2 ( e ^{2} - 1 ) ^{\frac{5}{2}}} (x - 1)^{2} - \frac{- 20 e ^{6} + 88 e ^{4} + 172 e ^{2}}{6 ( e ^{2} - 1 ) ^{\frac{7}{2}}} (x - 1)^{3} - \frac{8 e ^{8} + 288 e ^{6} - 2520 e ^{4} - 1136 e ^{2}}{24 ( e ^{2} - 1 ) ^{\frac{9}{2}}} (x - 1)^{4} + \dots