麦克劳林 sin((xpi)/4)

解答

麦克劳林

sin (\frac{x π}{4})

\frac{π}{4} x - \frac{π ^{3}}{384} x^{3} + \frac{π ^{5}}{122880} x^{5} - \frac{π ^{7}}{82575360} x^{7} + \frac{π ^{9}}{95126814720} x^{9} + \dots

求解步骤

应用麦克劳林公式

= 0 + \frac{\frac{d}{d x} ( sin ( \frac{x π}{4} ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( sin ( \frac{x π}{4} ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( sin ( \frac{x π}{4} ) ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 0 + \frac{\frac{π}{4}}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- \frac{π ^{3}}{64}}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{\frac{π ^{5}}{1024}}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- \frac{π ^{7}}{16384}}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{\frac{π ^{9}}{262144}}{9 !} x^{9} + \dots

整理后得

= \frac{π}{4} x - \frac{π ^{3}}{384} x^{3} + \frac{π ^{5}}{122880} x^{5} - \frac{π ^{7}}{82575360} x^{7} + \frac{π ^{9}}{95126814720} x^{9} + \dots

x \to \infty lim (\frac{1}{0})

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