1.04d Binomial expansion validity: convergence conditions

5 Find the first three terms in the binomial expansion of \(\sqrt [ 3 ] { 1 + 3 x }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.

6

1. Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where \(A , B\) and \(C\) are constants, find \(B\) and \(C\), and show that \(A = 0\).
2. Given that \(x\) is sufficiently small, find the first three terms of the binomial expansions of \(( 1 + x ) ^ { - 2 }\) and \(( 1 - 4 x ) ^ { - 1 }\). Hence find the first three terms of the expansion of \(\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }\).

7 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.

2 Given the binomial expansion \(( 1 + q x ) ^ { p } = 1 - x + 2 x ^ { 2 } + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]

3 Find the first three terms in the binomial expansion of \(\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]

4 Find the first three terms in the binomial expansion of \(\frac { 1 + 2 x } { ( 1 - 2 x ) ^ { 2 } }\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid.
[0pt] [7]

5 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.

6

1. Find the first three terms in the binomial expansion of \(\frac { 1 } { \sqrt { 1 - 2 x } }\). State the set of values of \(x\) for which the expansion is valid.
2. Hence find the first three terms in the series expansion of \(\frac { 1 + 2 x } { \sqrt { 1 - 2 x } }\).

7

1. Find the first three non-zero terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 2 } } }\) for \(| x | < 2\). [4]
2. Use this result to find an approximation for \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to
   4 significant figures.
3. Given that \(\int \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c\), evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to 4 significant figures.

6 Find the first three terms in the binomial expansion of \(\overline { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.

9

1. Find the first three non-zero terms of the binomial series expansion of \(\frac { 1 } { \sqrt { 1 + 4 x ^ { 2 } } }\), and state the set of values of \(x\) for which the expansion is valid.
2. Hence find the first three non-zero terms of the series expansion of \(\frac { 1 - x ^ { 2 } } { \sqrt { 1 + 4 x ^ { 2 } } }\).

10 Two students are trying to evaluate the integral \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\).
Sarah uses the trapezium rule with 2 strips, and starts by constructing the following table.

1. Complete the calculation, giving your answer to 3 significant figures. Anish uses a binomial approximation for \(\sqrt { 1 + \mathrm { e } ^ { - x } }\) and then integrates this.
2. Show that, provided \(\mathrm { e } ^ { - x }\) is suitably small, \(\left( 1 + \mathrm { e } ^ { - x } \right) ^ { \frac { 1 } { 2 } } \approx 1 + \frac { 1 } { 2 } \mathrm { e } ^ { - x } \quad \frac { 1 } { 8 } \mathrm { e } ^ { - 2 x }\).
3. Use this result to evaluate \(\int _ { 1 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { - x } } \mathrm {~d} x\) approximately, giving your answer to 3 significant figures.

3.(a)(i)Write down the binomial series expansion of $$\left( 1 + \frac { 2 } { n } \right) ^ { n } \quad n \in \mathbb { N } , n > 2$$ in powers of \(\left( \frac { 2 } { n } \right)\) up to and including the term in \(\left( \frac { 2 } { n } \right) ^ { 3 }\) (ii)Hence prove that,for \(n \in \mathbb { N } , n \geqslant 3\) $$\left( 1 + \frac { 2 } { n } \right) ^ { n } \geqslant \frac { 19 } { 3 } - \frac { 6 } { n }$$ (b)Use the binomial series expansion of \(\left( 1 - \frac { x } { 4 } \right) ^ { \frac { 1 } { 2 } }\) to show that \(\sqrt { 3 } < \frac { 7 } { 4 }\) $$\mathrm { f } ( x ) = \left( 1 + \frac { 2 } { x } \right) ^ { x } - 3 ^ { \frac { x } { 6 } } \quad x \in \mathbb { R } , x > 0$$ Given that the function \(\mathrm { f } ( x )\) is continuous and that \(\sqrt [ 6 ] { 3 } > \frac { 6 } { 5 }\) (c)prove that \(\mathrm { f } ( x ) = 0\) has a root in the interval[9,10]

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.In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) are equal and non-zero.

1. Find the possible values of \(n\) .
   (4)
2. State,giving a reason,which value of \(n\) gives a valid expansion when \(x = \frac { 1 } { 2 }\) (2)

3. (a) Find the binomial expansion of $$( 1 + a x ) ^ { - 3 } , \quad | a x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(a\). $$f ( x ) = \frac { 2 + 3 x } { ( 1 + a x ) ^ { 3 } } , \quad | a x | < 1$$ In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is 3
Given that \(a < 0\) (b) find the value of the constant \(a\),
(c) find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

2

1. Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

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.

3

1. Expand \(( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. Hence find the expansion of \(\frac { ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } } { ( 1 + x ) ^ { 3 } }\) as a series in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in part (ii) is valid.

5

1. Expand \(( 1 + x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
2. (a) Hence, or otherwise, expand \(( 8 + 16 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
   (b) State the set of values of \(x\) for which the expansion in part (ii) (a) is valid.

4

1. Expand \(( 1 - 4 x ) ^ { \frac { 1 } { 4 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. The term of lowest degree in the expansion of $$( 1 + a x ) \left( 1 + b x ^ { 2 } \right) ^ { 7 } - ( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$$ in ascending powers of \(x\) is the term in \(x ^ { 3 }\). Find the values of the constants \(a\) and \(b\).

2 Find the first three terms in the expansion of \(( 9 - 16 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\), and state the set of values for which this expansion is valid.

3

1. Expand \(( a + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
2. When \(( 1 - x ) ( a + x ) ^ { - 2 }\) is expanded, the coefficient of \(x ^ { 2 }\) is 0 . Find the value of \(a\).

3

1. Expand \(\frac { 1 + x ^ { 2 } } { \sqrt { 1 + 4 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
2. State the set of values of \(x\) for which this expansion is valid.