What is taylor arctan(2x) ?

The answer to taylor arctan(2x) is 2x-8/3 x^3+32/5 x^5-128/7 x^7+512/9 x^9+\ldots

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taylor arctan(2x)

taylor

arctan (2 x)

2 x - \frac{8}{3} x^{3} + \frac{32}{5} x^{5} - \frac{128}{7} x^{7} + \frac{512}{9} x^{9} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{2}{1 !} x + \frac{0}{2 !} x^{2} + \frac{- 16}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{768}{5 !} x^{5} + \frac{0}{6 !} x^{6} + \frac{- 92160}{7 !} x^{7} + \frac{0}{8 !} x^{8} + \frac{20643840}{9 !} x^{9} + \dots

Refine

= 2 x - \frac{8}{3} x^{3} + \frac{32}{5} x^{5} - \frac{128}{7} x^{7} + \frac{512}{9} x^{9} + \dots

y^{^{'}} = x y^{3}

\int 9 - 4 x d x

x \to - 1 lim (x^{3} - x^{2} - 5 x + 3)

t an g e n t y = x x

y^{^{''}} - (\frac{2}{x}) y^{^{'}} - (\frac{7}{x ^{2}}) y = 0

What is taylor arctan(2x) ?
The answer to taylor arctan(2x) is 2x-8/3 x^3+32/5 x^5-128/7 x^7+512/9 x^9+\ldots