麦克劳林 1/((1-x)^2)

解答

麦克劳林

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

1 + 2 x + 3 x^{2} + 4 x^{3} + 5 x^{4} + \dots

求解步骤

应用麦克劳林公式

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

求导数

= 1 + \frac{2}{1 !} x + \frac{6}{2 !} x^{2} + \frac{24}{3 !} x^{3} + \frac{120}{4 !} x^{4} + \dots

整理后得

= 1 + 2 x + 3 x^{2} + 4 x^{3} + 5 x^{4} + \dots

\frac{\partial}{\partial x} (x^{2} + x y + 5 y)

x \to 2 lim (6 x^{2})

\frac{d}{d x} (\frac{x ^{2} + x}{2 x + 5})

x \to - \infty lim (\frac{x}{- x})

t an g e n t 12 x^{\frac{1}{2}}