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]
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Answer:(i) \(f(0)=0,f'(0)=2,f''(0)=-4,f'''(0)=14\); \(2x-2x^2+\frac73x^3\). (ii) \(a=-1,n=2\), third term \(-x^3/3\).