Maclaurin's series

Question code: 263

Question 0703

(i)

By successively differentiating ln⁡(1+x){\ln ( 1 + x )}, find the Maclaurin's series for ln⁡(1+x){\ln ( 1 + x )}, up to and including the term in x3.{x^3.}
[4]

(ii)

Obtain the expansion of ln⁡(6−x)+ln⁡(1+3x2){\ln ( 6 - x ) + \ln ( 1 + 3 x^2 )} up to and including the term in x3.{x^3.}
[5]

(iii)

Find the set of values of x{x} for which the expansion in part (ii) is valid.
[2]
Answer

(i)

x−12x2+13x3+…{x - \frac{1}{2} x^2 + \frac{1}{3} x^3 + \ldots}

(ii)

ln⁡6−16x+21572x2−1648x3+…{\ln 6 - \frac{1}{6} x + \frac{215}{72} x^2 - \frac{1}{648} x^3 + \ldots}

(iii)

−133≤x≤133.{- \frac{1}{3} \sqrt{3} \leq x \leq \frac{1}{3} \sqrt{3}.}
Question code: 263