N2013 P2 Q3

Given that \(f(x)=\ln(1+2\sin x)\), find \(f(0)\), \(f'(0)\), \(f''(0)\) and \(f'''(0)\). Hence write down the first three non-zero terms in the Maclaurin series for \(f(x)\).[7]

The first two non-zero terms in the Maclaurin series for \(f(x)\) are equal to the first two non-zero terms in the series expansion of \(\mathrm e^{ax}\sin nx\). Using appropriate expansions from the List of Formulae (MF15), find the constants \(a\) and \(n\). Hence find the third non-zero term of the series expansion of \(\mathrm e^{ax}\sin nx\) for these values of \(a\) and \(n\).[5]