Expansion with algebraic manipulation

5

1. Simplify \(( \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) ) ( \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) )\), showing your working, and deduce that $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) } = \frac { \sqrt { } ( 1 + x ) - \sqrt { } ( 1 - x ) } { 2 x }$$
2. Using this result, or otherwise, obtain the expansion of $$\frac { 1 } { \sqrt { } ( 1 + x ) + \sqrt { } ( 1 - x ) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form. Given that $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { n } \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { - \frac { 3 } { 2 } } = \left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } }$$ (b) write down the value of \(n\).
(c) Hence, or otherwise, find the first 3 terms of the binomial expansion, in ascending powers of \(x\), of $$\left( \frac { 1 } { 4 } - \frac { 1 } { 2 } x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 2 }$$ giving each term in simplest form.

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）Write down the binomial expansion of \(\frac { 1 } { ( 1 - y ) ^ { 2 } } , | y | < 1\) ，in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) ．
（b）Hence，or otherwise，show that $$\frac { 1 } { 4 } \operatorname { cosec } ^ { 4 } \left( \frac { \theta } { 2 } \right) = 1 + 2 \cos \theta + 3 \cos ^ { 2 } \theta + 4 \cos ^ { 3 } \theta + \ldots + ( r + 1 ) \cos ^ { r } \theta + \ldots$$ and state the values of \(\theta\) for which this result is not valid．
（4）
Find
（c） $$\begin{aligned} & 1 + \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } + \frac { 4 } { 2 ^ { 3 } } + \ldots + \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots
& 1 - \frac { 2 } { 2 } + \frac { 3 } { 2 ^ { 2 } } - \frac { 4 } { 2 ^ { 3 } } + \ldots + ( - 1 ) ^ { r } \frac { ( r + 1 ) } { 2 ^ { r } } + \ldots \end{aligned}$$ （d）

1. Given that

$$( 1 + x ) ^ { n } = 1 + \sum _ { r = 1 } ^ { \infty } \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { 1 \times 2 \times \ldots \times r } x ^ { r } \quad ( | x | < 1 , x \in \mathbb { R } , n \in \mathbb { R } )$$

1. show that $$( 1 - x ) ^ { - \frac { 1 } { 2 } } = \sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \left( \frac { x } { 4 } \right) ^ { r }$$
2. show that \(\left( 9 - 4 x ^ { 2 } \right) ^ { - \frac { 1 } { 2 } }\) can be written in the form \(\sum _ { r = 0 } ^ { \infty } \binom { 2 r } { r } \frac { x ^ { 2 r } } { 3 ^ { q } }\) and give \(q\) in terms of \(r\).
3. Find \(\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r } { 9 } \times \left( \frac { x } { 3 } \right) ^ { 2 r - 1 }\)
4. Hence find the exact value of $$\sum _ { r = 1 } ^ { \infty } \binom { 2 r } { r } \times \frac { 2 r \sqrt { } 5 } { 9 } \times \frac { 1 } { 5 ^ { r } }$$ giving your answer as a rational number.

12.

1. Show that the first two terms of the binomial expansion of \(\sqrt { 4 - 2 x ^ { 2 } }\) are $$2 - \frac { x ^ { 2 } } { 2 }$$
2. State the range of values of \(x\) for which the expansion found in part (a) is valid.
3. Hence, find an approximation for $$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } d x$$ giving your answer to five decimal places.
   Fully justify your answer.
4. A student decides to use this method to find an approximation for $$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } d x$$ Explain why this may not be a suitable method.

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1. Show that the first two terms of the binomial expansion of \(\sqrt { 4 - 2 x ^ { 2 } }\) are $$2 - \frac { x ^ { 2 } } { 2 }$$ 9
2. State the range of values of \(x\) for which the expansion found in part (a) is valid.
   [0pt] [2 marks]
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3. Hence, find an approximation for $$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } \mathrm {~d} x$$ giving your answer to five decimal places.
   Fully justify your answer.
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4. A student decides to use this method to find an approximation for $$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } \mathrm {~d} x$$ Explain why this may not be a suitable method.