What is taylor x/(1+3x^2) ?

The answer to taylor x/(1+3x^2) is x-3x^3+9x^5-27x^7+81x^9+\ldots

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taylor x/(1+3x^2)

taylor

\frac{x}{1 + 3 x ^{2}}

x - 3 x^{3} + 9 x^{5} - 27 x^{7} + 81 x^{9} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 18}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{1080}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 136080}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{29393280}{9 !} x^{9} + \dots

Refine

= x - 3 x^{3} + 9 x^{5} - 27 x^{7} + 81 x^{9} + \dots

What is taylor x/(1+3x^2) ?
The answer to taylor x/(1+3x^2) is x-3x^3+9x^5-27x^7+81x^9+\ldots