What is taylor ln(1+2x) ?

The answer to taylor ln(1+2x) is 2x-2x^2+8/3 x^3-4x^4+32/5 x^5+\ldots

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taylor ln(1+2x)

taylor

ln (1 + 2 x)

2 x - 2 x^{2} + \frac{8}{3} x^{3} - 4 x^{4} + \frac{32}{5} x^{5} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{2}{1 !} x + \frac{- 4}{2 !} x^{2} + \frac{16}{3 !} x^{3} + \frac{- 96}{4 !} x^{4} + \frac{768}{5 !} x^{5} + \dots

Refine

= 2 x - 2 x^{2} + \frac{8}{3} x^{3} - 4 x^{4} + \frac{32}{5} x^{5} + \dots

\frac{d}{d x} (6 x^{2} - 6 x - 12)

(ln (x))^{3}

\int \frac{2 t}{1 - 4 t ^{4}} d t

\frac{d x}{d t} = - x

\frac{d}{d x} (1.5)

What is taylor ln(1+2x) ?
The answer to taylor ln(1+2x) is 2x-2x^2+8/3 x^3-4x^4+32/5 x^5+\ldots