泰勒 (x-1)ln(x)

解答

泰勒

(x - 1) ln (x)

(x - 1)^{2} - \frac{1}{2} (x - 1)^{3} + \frac{1}{3} (x - 1)^{4} - \frac{1}{4} (x - 1)^{5} + \frac{1}{5} (x - 1)^{6} + \dots

求解步骤

应用泰勒公式

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

求导数

= 0 + \frac{0}{1 !} (x - 1) + \frac{2}{2 !} (x - 1)^{2} + \frac{- 3}{3 !} (x - 1)^{3} + \frac{8}{4 !} (x - 1)^{4} + \frac{- 30}{5 !} (x - 1)^{5} + \frac{144}{6 !} (x - 1)^{6} + \dots

整理后得

= (x - 1)^{2} - \frac{1}{2} (x - 1)^{3} + \frac{1}{3} (x - 1)^{4} - \frac{1}{4} (x - 1)^{5} + \frac{1}{5} (x - 1)^{6} + \dots

\frac{\partial}{\partial x} (\frac{1}{3 x + 2})

x \to 2 - lim (g (x))

\int \frac{x}{2 x ^{2} + 1} d x

x \to \infty lim (f (x) x)

\int 8 ln (x) d x