Use binomial with exponential series

6 The function f is defined by $\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } ( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }$.

1. Use the series expansion for $\mathrm { e } ^ { x }$ to write down the first four terms in the series expansion of $\mathrm { e } ^ { 2 x }$.
2. Use the binomial series expansion of $( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }$ and your answer to part (a)(i) to show that the first three non-zero terms in the series expansion of $\mathrm { f } ( x )$ are $1 + 3 x ^ { 2 } - 6 x ^ { 3 }$.
1. Given that $y = \ln ( 1 + 2 \sin x )$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
2. By using Maclaurin's theorem, show that, for small values of $x$, $$\ln ( 1 + 2 \sin x ) \approx 2 x - 2 x ^ { 2 }$$
Find $$\lim _ { x \rightarrow 0 } \frac { 1 - \mathrm { f } ( x ) } { x \ln ( 1 + 2 \sin x ) }$$

2

Write down the expansion of $\mathrm { e } ^ { 3 x }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
Hence, or otherwise, find the term in $x ^ { 2 }$ in the expansion, in ascending powers of $x$, of $\mathrm { e } ^ { 3 x } ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }$.
(4 marks)

6 The function f is defined by $\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }$.

1. Find f'''(x).
2. Using Maclaurin's theorem, show that, for small values of $x$, $$\mathrm { f } ( x ) \approx 1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
Use the expansion of $\mathrm { e } ^ { x }$ together with the result in part (a)(ii) to show that, for small values of $x$, $$\mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \approx 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where $k$ is a rational number to be found.
Write down the first four terms in the expansion, in ascending powers of $x$, of $\mathrm { e } ^ { 2 x }$.
Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } - \mathrm { e } ^ { 2 x } } { 1 - \cos x }$$ (4 marks)