Question 0813

Given that ${f(x)=\ln(1+4 \sin 2 x),}$ find ${f(0),}$ ${f'(0),}$ ${f''(0)}$ and ${f'''(0).}$ Hence write down the first three non-zero terms in the Maclaurin series for ${f(x).}$

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 ${\mathrm{e}^{ax} \sin(nx)}$ for these values of ${a}$ and ${n.}$