泰勒 x/(10-9x)

解答

泰勒

\frac{x}{10 - 9 x}

\frac{1}{10} x + \frac{9}{100} x^{2} + \frac{81}{1000} x^{3} + \frac{729}{10000} x^{4} + \frac{6561}{100000} x^{5} + \dots

求解步骤

应用泰勒公式

= 0 + \frac{\frac{d}{d x} ( \frac{x}{10 - 9 x} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{x}{10 - 9 x} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{x}{10 - 9 x} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 0 + \frac{\frac{1}{10}}{1 !} x + \frac{\frac{9}{50}}{2 !} x^{2} + \frac{\frac{243}{500}}{3 !} x^{3} + \frac{\frac{2187}{1250}}{4 !} x^{4} + \frac{\frac{19683}{2500}}{5 !} x^{5} + \dots

整理后得

= \frac{1}{10} x + \frac{9}{100} x^{2} + \frac{81}{1000} x^{3} + \frac{729}{10000} x^{4} + \frac{6561}{100000} x^{5} + \dots

\int (25 - x^{2})^{- \frac{3}{2}} d x

\frac{d}{d x} (1 0^{2 x})

\frac{d b}{d b}

d er i v a t i v e (3 x^{4} + 3 x)^{2}

n = 0 \sum \infty \frac{3}{7 ^{n}}