麦克劳林 cos^2(x)

解答

麦克劳林

cos^{2} (x)

1 - x^{2} + \frac{1}{3} x^{4} - \frac{2}{45} x^{6} + \frac{1}{315} x^{8} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= 1 + \frac{0}{1 !} x + \frac{- 2}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{8}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 32}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{128}{8 !} x^{8} + \dots

整理后得

= 1 - x^{2} + \frac{1}{3} x^{4} - \frac{2}{45} x^{6} + \frac{1}{315} x^{8} + \dots

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