泰勒 1/(x-1),18

解答

泰勒

\frac{1}{x - 1}, 18

\frac{1}{17} - \frac{1}{289} (x - 18) + \frac{1}{4913} (x - 18)^{2} - \frac{1}{83521} (x - 18)^{3} + \frac{1}{1419857} (x - 18)^{4} + \dots

求解步骤

应用泰勒公式

= \frac{1}{17} + \frac{\frac{d}{d x} ( \frac{1}{x - 1} ) ( 18 )}{1 !} (x - 18) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{x - 1} ) ( 18 )}{2 !} (x - 18)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{x - 1} ) ( 18 )}{3 !} (x - 18)^{3} + \dots

求导数

= \frac{1}{17} + \frac{- \frac{1}{289}}{1 !} (x - 18) + \frac{\frac{2}{4913}}{2 !} (x - 18)^{2} + \frac{- \frac{6}{83521}}{3 !} (x - 18)^{3} + \frac{\frac{24}{1419857}}{4 !} (x - 18)^{4} + \dots

整理后得

= \frac{1}{17} - \frac{1}{289} (x - 18) + \frac{1}{4913} (x - 18)^{2} - \frac{1}{83521} (x - 18)^{3} + \frac{1}{1419857} (x - 18)^{4} + \dots

a re a sin (x), cos (x)

x \to 5 - lim (25 - x^{2})

t an g e n t y = x, (81, 9)

x \to 0 lim (sin (x))

\int \frac{1}{x y} d x