Factor and rescale

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．

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1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

5

1. 1. Obtain the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
      (2 marks)
   2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of \(x\).
2. Obtain the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).
3. Given that \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) can be written in the form \(\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }\), find the values of \(A , B\) and \(C\).
4. Hence find the binomial expansion of \(\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }\) up to and including the term in \(x ^ { 2 }\).

9

1. 1. Find the binomial expansion of \(( 1 + 3 x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\)
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2. (ii) Show that the first three terms in the binomial expansion of $$\frac { 1 } { 2 - 3 x }$$ form a geometric sequence and state the common ratio.
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3. It is given that $$\frac { 36 x } { ( 1 + 3 x ) ( 2 - 3 x ) } \equiv \frac { P } { ( 2 - 3 x ) } + \frac { Q } { ( 1 + 3 x ) }$$ where \(P\) and \(Q\) are integers. Find the value of \(P\) and the value of \(Q\)
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4. 1. Using your answers to parts (a) and (b), find the binomial expansion of $$\frac { 12 x } { ( 1 + 3 x ) ( 2 - 3 x ) }$$ up to and including the term in \(x ^ { 2 }\)
      [0pt] [2 marks]
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5. (ii) Find the range of values of \(x\) for which the binomial expansion of $$\frac { 12 x } { ( 1 + 3 x ) ( 2 - 3 x ) }$$ is valid.