泰勒 1/(sqrt(3-2x)),1

解答

泰勒

\frac{1}{3 - 2 x}, 1

1 + (x - 1) + \frac{3}{2} (x - 1)^{2} + \frac{5}{2} (x - 1)^{3} + \frac{35}{8} (x - 1)^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= 1 + \frac{1}{1 !} (x - 1) + \frac{3}{2 !} (x - 1)^{2} + \frac{15}{3 !} (x - 1)^{3} + \frac{105}{4 !} (x - 1)^{4} + \dots

整理后得

= 1 + (x - 1) + \frac{3}{2} (x - 1)^{2} + \frac{5}{2} (x - 1)^{3} + \frac{35}{8} (x - 1)^{4} + \dots

d er i v a t i v e f (x) = x^{2} - 3 x

t an g e n t \frac{1}{3 + 2 x}

t a y l or \frac{1}{1 - x}, 0

x \to 0 + lim (ln^{2} (x))

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