泰勒 ln(2/(2-x))

解答

泰勒

ln (\frac{2}{2 - x})

\frac{1}{2} x + \frac{1}{8} x^{2} + \frac{1}{24} x^{3} + \frac{1}{64} x^{4} + \frac{1}{160} x^{5} + \dots

求解步骤

应用泰勒公式

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

求导数

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

整理后得

= \frac{1}{2} x + \frac{1}{8} x^{2} + \frac{1}{24} x^{3} + \frac{1}{64} x^{4} + \frac{1}{160} x^{5} + \dots

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

ma c l a u r in \frac{1}{1 + x ^{2}}, x = 2

x \to - 9 - lim (\frac{x + 10}{x + 9})

\frac{\partial}{\partial x} (x^{n} y \frac{1}{x ^{2} + y ^{2}})

\int (1 - x^{3})^{2} d x