1.04d Binomial expansion validity: convergence conditions

2

1. Expand \(\frac { 1 } { \sqrt { } ( 1 - 4 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 1 + 2 x } { \sqrt { } ( 4 - 16 x ) }\).

3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

1 Expand \(\frac { 1 } { \sqrt { } ( 4 + 3 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

3 Show that, for small values of \(x ^ { 2 }\), $$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$ where the value of the constant \(k\) is to be determined.

2 Expand \(( 1 + x ) \sqrt { } ( 1 - 2 x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

4 When \(( 1 + a x ) ^ { - 2 }\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 3 }\) are equal.

1. Find the exact value of \(a\).
2. When \(a\) has this value, obtain the expansion up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2 Expand \(( 2 - x ) ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.

2

1. Expand \(( 2 - 3 x ) ^ { - 2 }\) 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.

2

1. Expand \(\left( 2 - x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

2

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 x - 5 ) \sqrt { 4 - x }\).
2. State the set of values of \(x\) for which the expansion in part (a) is valid.

2

1. Expand \(\sqrt [ 3 ] { 1 + 6 x }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
2. State the set of values of \(x\) for which the expansion is valid.

6. Given that the binomial expansion, in ascending powers of \(x\), of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is \(\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots\)

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the coefficient of \(x ^ { 6 }\)

1. (a) Use the binomial expansion, in ascending powers of \(x\), of \(\frac { 1 } { \sqrt { } ( 1 - 2 x ) }\) to show that

$$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } \approx 2 + 5 x + 6 x ^ { 2 } , \quad | x | < 0.5$$ (b) Substitute \(x = \frac { 1 } { 20 }\) into $$\frac { 2 + 3 x } { \sqrt { } ( 1 - 2 x ) } = 2 + 5 x + 6 x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 10 }\) Give your answer as a fraction in its simplest form.

2. $$f ( x ) = ( 125 - 5 x ) ^ { \frac { 2 } { 3 } } \quad | x | < 25$$

1. Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving the coefficient of \(x\) and the coefficient of \(x ^ { 2 }\) as simplified fractions.
2. Use your expansion to find an approximate value for \(120 ^ { \frac { 2 } { 3 } }\), stating the value of \(x\) which you have used and showing your working. Give your answer to 5 decimal places.

4. (a) Find the binomial expansion of $$( 1 + p x ) ^ { - 4 } , \quad | p x | < 1$$ 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 \(p\). $$f ( x ) = \frac { 3 + 4 x } { ( 1 + p x ) ^ { 4 } } \quad | p x | < 1$$ where \(p\) is a positive constant. In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\).
(b) Find the value of \(p\).
(c) Hence find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.

1. (a) Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction.
(b) Hence or otherwise, find the coefficient of \(x ^ { 2 }\) in the series expansion of

1. \(\frac { 1 } { ( 2 + 6 x ) ^ { 3 } } \quad | x | < \frac { 1 } { 3 }\)
2. \(\frac { 4 - x } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }\)

1. (a) Use the binomial series to expand

$$\frac { 1 } { ( 2 - 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. $$f ( x ) = \frac { 4 + k x } { ( 2 - 3 x ) ^ { 3 } } \quad \text { where } k \text { is a constant and } | x | < \frac { 2 } { 3 }$$ Given that the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), is $$\frac { 1 } { 2 } + A x + \frac { 81 } { 16 } x ^ { 2 } + \cdots$$ where \(A\) is a constant,
(b) find the value of \(k\),
(c) find the value of \(A\).

6. (a) Use binomial expansions to show that, for \(| x | < \frac { 1 } { 2 }\) (b) Find the exact value of \(\sqrt { \frac { 1 + 2 x } { 1 - x } }\) when \(x = \frac { 1 } { 10 }\) Give your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into the expansion given in part (a) and hence find an approximate value for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. $$\sqrt { \frac { 1 + 2 x } { 1 - x } } \approx 1 + \frac { 3 } { 2 } x + \frac { 3 } { 8 } x ^ { 2 }$$

4. (a) Use the binomial theorem to expand $$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\) Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.

2. (a) Find, in ascending powers of \(x\), the first three non-zero terms of the binomial series expansion of $$\sqrt [ 3 ] { 1 + 4 x ^ { 3 } } \quad | x | < \frac { 1 } { \sqrt [ 3 ] { 4 } }$$ giving each coefficient as a simplified fraction.
(b) Use the expansion from part (a) with \(x = \frac { 1 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 31 }\) (3)

1. $$f ( x ) = \frac { 5 x + 10 } { ( 1 - x ) ( 2 + 3 x ) }$$

1. Write \(\mathrm { f } ( x )\) in partial fraction form.
2. 1. Hence find, in ascending powers of \(x\) up to and including the terms in \(x ^ { 2 }\), the binomial series expansion of \(\mathrm { f } ( x )\). Give each coefficient as a simplified fraction.
   2. Find the range of values of \(x\) for which this expansion is valid.

2. (a) Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 1 } { 3 } } , \quad | x | < \frac { 8 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each term as a simplified fraction.
(b) Use your expansion, with a suitable value of \(x\), to obtain an approximation to \(\sqrt [ 3 ] { } ( 7.7 )\). Give your answer to 7 decimal places.

1. (a) Use the binomial theorem to expand

$$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction. $$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$ In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\). Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.

3. (a) Expand $$\frac { 1 } { ( 2 - 5 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction. Given that the binomial expansion of \(\frac { 2 + k x } { ( 2 - 5 x ) ^ { 2 } } , | x | < \frac { 2 } { 5 }\), is $$\frac { 1 } { 2 } + \frac { 7 } { 4 } x + A x ^ { 2 } + \ldots$$ (b) find the value of the constant \(k\),
(c) find the value of the constant \(A\).

1. Use the binomial theorem to expand

$$\sqrt { } ( 4 - 9 x ) , \quad | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.