麦克劳林 ln(1/(1-x))

解答

麦克劳林

ln (\frac{1}{1 - x})

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

求解步骤

应用麦克劳林公式

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

求导数

= 0 + \frac{1}{1 !} x + \frac{1}{2 !} x^{2} + \frac{2}{3 !} x^{3} + \frac{6}{4 !} x^{4} + \frac{24}{5 !} x^{5} + \dots

整理后得

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

(x^{2} y^{3} - \frac{1}{1 + 9 x ^{2}}) d x + (x^{3} y^{2}) d y = 0

\frac{d}{d x} (sin^{2} (x) - cos^{2} (x))

x \to 5 lim (\frac{3}{x - 5})

\frac{d}{d x} (∣ x ∣^{x})

\int \frac{1}{x - x - 2} d x