Question 0813

Maclaurin Series

2013 Paper 2 Question 3 Variant

Question

(i)

Given that

{f(x)=\ln(1+2 \sin 3 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]

(ii)

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.}

[5]