Match series to function form

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

1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
2. It is given that the first non-zero term in the expansion of $$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$ in ascending powers of \(x\) is \(q x ^ { 5 }\).
   Find the values of the rational numbers \(p\) and \(q\).

8 You are given that \(\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }\) where \(a\) and \(b\) are positive constants. The first three terms in the Maclaurin expansion of \(\mathrm { f } ( x )\) are \(1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }\).

1. Find the value of \(a\) and the value of \(b\).
2. Explain if there is any restriction on the value of \(x\) in order for the expansion to be valid.

Which of the following functions has the fourth term \(-\frac{1}{720}x^6\) in its Maclaurin series expansion? Circle your answer. [1 mark] \(\sin x\) \(\qquad\) \(\cos x\) \(\qquad\) \(e^x\) \(\qquad\) \(\ln(1 + x)\)

The first two non-zero terms of the Maclaurin series expansion of \(f(x)\) are \(x\) and \(-\frac{1}{2}x^3\) Which one of the following could be \(f(x)\)? Circle your answer. [1 mark] \(xe^{\frac{1}{2}x^2}\) \quad \(\frac{1}{2}\sin 2x\) \quad \(x \cos x\) \quad \((1 + x^3)^{-\frac{1}{2}}\)