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

The answer to taylor (2x)/((1+x^2)^2) is 2x-4x^3+6x^5-8x^7+10x^9+\ldots

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taylor (2x)/((1+x^2)^2)

taylor

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

2 x - 4 x^{3} + 6 x^{5} - 8 x^{7} + 10 x^{9} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{2}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 24}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{720}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 40320}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{3628800}{9 !} x^{9} + \dots

Refine

= 2 x - 4 x^{3} + 6 x^{5} - 8 x^{7} + 10 x^{9} + \dots

\frac{d}{d x} (x^{3} + 5 x + 7)

e^{- t} + 2 e^{- 2 t} + t e^{- 3 t}

\int \frac{5 cosh ( x )}{x} d x

n = 0 \sum \infty \frac{n}{2 n ^{2} + 1}

x \to 4 lim (\frac{7}{x - 5})

What is taylor (2x)/((1+x^2)^2) ?
The answer to taylor (2x)/((1+x^2)^2) is 2x-4x^3+6x^5-8x^7+10x^9+\ldots