Challenging +1.8 This AEA question requires recognizing that the binomial expansion of (1-y)^{-2} relates to a series with variable coefficients and denominators, then manipulating it to identify the closed form. Part (b) demands pattern recognition and algebraic insight beyond standard textbook exercises, while part (c) requires understanding convergence conditions. The multi-step reasoning and non-routine connection between parts elevates this above typical A-level questions.
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)Hence,or otherwise,show that
$$1 + \frac { 2 x } { 1 + x } + \frac { 3 x ^ { 2 } } { ( 1 + x ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( 1 + x ) ^ { r - 1 } } + \ldots$$
can be written in the form \(( a + x ) ^ { n }\) .Write down the values of the integers \(a\) and \(n\) .
(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)Hence,or otherwise,show that
$$1 + \frac { 2 x } { 1 + x } + \frac { 3 x ^ { 2 } } { ( 1 + x ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( 1 + x ) ^ { r - 1 } } + \ldots$$
can be written in the form $( a + x ) ^ { n }$ .Write down the values of the integers $a$ and $n$ .\\
(c)Find the set of values of $x$ for which the series in part(b)is convergent.
\hfill \mbox{\textit{Edexcel AEA 2006 Q1 [8]}}