Standard +0.8 This is a multi-part Further Maths question requiring binomial expansion, series manipulation (including division of series), Maclaurin's theorem, and limit evaluation using series. While each individual technique is standard for FP3, the combination of methods and the algebraic manipulation needed (especially dividing series to get sec x) elevates this above routine. The limit in part (c) requires careful substitution and simplification of multiple series expansions.
Write down the first three terms of the binomial expansion of \(( 1 + y ) ^ { - 1 }\), in ascending powers of \(y\).
By using the expansion
$$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$
and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\sec x\) are
$$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\tan x\) are
$$x + \frac { x ^ { 3 } } { 3 }$$
Hence find \(\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)\).
7
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the first three terms of the binomial expansion of $( 1 + y ) ^ { - 1 }$, in ascending powers of $y$.
\item By using the expansion
$$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$
and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\sec x$ are
$$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
\end{enumerate}\item By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\tan x$ are
$$x + \frac { x ^ { 3 } } { 3 }$$
\item Hence find $\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2006 Q7 [13]}}