Challenging +1.2 This is a multi-part question requiring binomial expansion with negative exponent (routine for AEA), pattern recognition to identify the series as a binomial expansion (moderate insight needed), and convergence analysis. While it requires several techniques and some algebraic manipulation, the steps are fairly standard for Further Maths students and the pattern recognition, though not trivial, follows from comparing coefficients systematically. Harder than average A-level but not exceptionally challenging for AEA.
1.(a)For \(| y | < 1\) ,write down the binomial series expansion of \(( 1 - y ) ^ { - 2 }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\)
(b)Show that when it is convergent,the series
$$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$
can be written in the form \(( 1 + a x ) ^ { n }\) ,where \(a\) and \(n\) are constants to be found.
(c)Find the set of values of \(x\) for which the series in part(b)is convergent.
1.(a)For $| y | < 1$ ,write down the binomial series expansion of $( 1 - y ) ^ { - 2 }$ in ascending powers of $y$ up to and including the term in $y ^ { 3 }$\\
(b)Show that when it is convergent,the series
$$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$
can be written in the form $( 1 + a x ) ^ { n }$ ,where $a$ and $n$ are constants to be found.\\
(c)Find the set of values of $x$ for which the series in part(b)is convergent.
\hfill \mbox{\textit{Edexcel AEA 2017 Q1 [8]}}