What is taylor (x/(2-x))^3 ?

The answer to taylor (x/(2-x))^3 is 1/8 x^3+3/16 x^4+3/16 x^5+5/32 x^6+15/128 x^7+\ldots

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taylor (x/(2-x))^3

taylor

(\frac{x}{2 - x})^{3}

\frac{1}{8} x^{3} + \frac{3}{16} x^{4} + \frac{3}{16} x^{5} + \frac{5}{32} x^{6} + \frac{15}{128} x^{7} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{0}{1 !} x + \frac{0}{2 !} x^{2} + \frac{\frac{3}{4}}{3 !} x^{3} + \frac{\frac{9}{2}}{4 !} x^{4} + \frac{\frac{45}{2}}{5 !} x^{5} + \frac{\frac{225}{2}}{6 !} x^{6} + \frac{\frac{4725}{8}}{7 !} x^{7} + \dots

Refine

= \frac{1}{8} x^{3} + \frac{3}{16} x^{4} + \frac{3}{16} x^{5} + \frac{5}{32} x^{6} + \frac{15}{128} x^{7} + \dots

(x y - x^{2} - 1) d x + (x^{2} + 1) d y = c

\frac{d}{d x} (arccot (x^{2} + 1))

\int e^{7 x + 2} d x

\frac{x ^{6} + e ^{x}}{x + 3}

\frac{d}{d x} (\frac{1}{x + y})

What is taylor (x/(2-x))^3 ?
The answer to taylor (x/(2-x))^3 is 1/8 x^3+3/16 x^4+3/16 x^5+5/32 x^6+15/128 x^7+\ldots