Write down the first three non-zero terms in the Maclaurin series for \(\ln(1+2x)\), where \(-\frac12<x\le\frac12\), simplifying the coefficients.[2]
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.[6]
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Answer:(i) \(2x-2x^2+\frac83x^3\) (ii) \(a=2,b=\frac53,c=-\frac35\), coefficient \(-\frac{104}{27}\).