Question 0815

Write down the first three non-zero terms in the Maclaurin series for ${\ln(1 - 2 x),}$ where ${- \frac{1}{2} \leq x < \frac{1}{2},}$ simplifying the coefficients.

It is given that the three terms found in part (i) are equal to the first three terms in the series expansion of ${ax(1+bx)^c}$ for small ${x.}$

Find the exact values of the constants ${a, b}$ and ${c}$ and use these values to find the coefficient of ${x^4}$ in the expansion of ${ax(1+bx)^c,}$ giving your answer as a simplified rational number.