Match two series without matching their cubic terms

(i) For $f(x)=\ln(1+2\sin x)$, find $f(0),f^{\prime}(0),f^{\prime\prime}(0),f^{\prime\prime\prime}(0)$ and hence its first three nonzero Maclaurin terms.7 marks

(ii) Its first two nonzero terms match those of $e^{ax}\sin(nx)$. Use appropriate MF27 expansions to find $a,n$ and then the third nonzero term of $e^{ax}\sin(nx)$.5 marks