What is taylor ln(1+6x) ?

The answer to taylor ln(1+6x) is 6x-18x^2+72x^3-324x^4+7776/5 x^5+\ldots

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taylor ln(1+6x)

taylor

ln (1 + 6 x)

6 x - 18 x^{2} + 72 x^{3} - 324 x^{4} + \frac{7776}{5} x^{5} + \dots

Solution steps

Apply the Taylor Formula

= 0 + \frac{\frac{d}{d x} ( ln ( 1 + 6 x ) ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ln ( 1 + 6 x ) ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ln ( 1 + 6 x ) ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 0 + \frac{6}{1 !} x + \frac{- 36}{2 !} x^{2} + \frac{432}{3 !} x^{3} + \frac{- 7776}{4 !} x^{4} + \frac{186624}{5 !} x^{5} + \dots

Refine

= 6 x - 18 x^{2} + 72 x^{3} - 324 x^{4} + \frac{7776}{5} x^{5} + \dots

x \to 2 lim (\frac{x}{ln ( x - 1 )})

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

\int \frac{x ^{2} + 4 x - 4}{x} d x

f (x) = 6 x^{3} - 7 x

What is taylor ln(1+6x) ?
The answer to taylor ln(1+6x) is 6x-18x^2+72x^3-324x^4+7776/5 x^5+\ldots