麦克劳林 4cos(x)

解答

麦克劳林

4 cos (x)

4 - 2 x^{2} + \frac{1}{6} x^{4} - \frac{1}{180} x^{6} + \frac{1}{10080} x^{8} + \dots

求解步骤

应用麦克劳林公式

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

求导数

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

整理后得

= 4 - 2 x^{2} + \frac{1}{6} x^{4} - \frac{1}{180} x^{6} + \frac{1}{10080} x^{8} + \dots

n = 1 \sum \infty (- 1)^{n} n^{- 2}

in v erse l a pl a ce \frac{1}{( s + 1 ) ^{2} ( s ^{2} + 1 )}

(x 4 - x^{2})^{^{'}}

n = 1 \sum \infty \frac{1 ^{n}}{n ^{7}}

x \to 0 lim (\frac{x ^{2}}{∣ x ∣})