Standard +0.8 This requires computing a Maclaurin series for a product (using product rule or the exponential series), extracting coefficients, then matching to binomial expansion coefficients and solving simultaneous equations. It's a multi-step Further Maths question requiring careful algebraic manipulation, but follows a standard template for series matching problems.
2 Given that the first three terms of the Maclaurin series for \(( 1 + \sin x ) \mathrm { e } ^ { 2 x }\) are identical to the first three terms of the binomial series for \(( 1 + a x ) ^ { n }\), find the values of the constants \(a\) and \(n\). (You may use appropriate results given in the List of Formulae (MF1).)
2 Given that the first three terms of the Maclaurin series for $( 1 + \sin x ) \mathrm { e } ^ { 2 x }$ are identical to the first three terms of the binomial series for $( 1 + a x ) ^ { n }$, find the values of the constants $a$ and $n$. (You may use appropriate results given in the List of Formulae (MF1).)
\hfill \mbox{\textit{OCR FP2 2010 Q2 [6]}}