What is taylor x/(1-x^2),0 ?

The answer to taylor x/(1-x^2),0 is x+x^3+x^5+x^7+x^9+\ldots

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taylor x/(1-x^2),0

taylor

\frac{x}{1 - x ^{2}}, 0

x + x^{3} + x^{5} + x^{7} + x^{9} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{6}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{120}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{5040}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{362880}{9 !} x^{9} + \dots

Refine

= x + x^{3} + x^{5} + x^{7} + x^{9} + \dots

What is taylor x/(1-x^2),0 ?
The answer to taylor x/(1-x^2),0 is x+x^3+x^5+x^7+x^9+\ldots