1.04c Extend binomial expansion: rational n, |x|<1

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.

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

$$\left( \frac { 1 } { 4 } - 5 x \right) ^ { \frac { 1 } { 2 } } \quad | x | < \frac { 1 } { 20 }$$ giving each coefficient in its simplest form. By substituting \(x = \frac { 1 } { 100 }\) into the answer for (a),
(b) find an approximation for \(\sqrt { 5 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.

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

$$\frac { 2 } { \sqrt { 9 - 2 x } } \quad | x | < \frac { 9 } { 2 }$$ giving each coefficient as a simplified fraction. By substituting \(x = 1\) into the answer to part (a),
(b) find an approximation for \(\sqrt { 7 }\), giving your answer to 4 decimal places.

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.

1. Find, in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), the binomial expansion of

$$( 1 - 4 x ) ^ { - 3 } \quad | x | < \frac { 1 } { 4 }$$ fully simplifying each term.

5. $$f ( x ) = \frac { 3 x ^ { 2 } + 16 } { ( 1 - 3 x ) ( 2 + x ) ^ { 2 } } = \frac { A } { ( 1 - 3 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 3 } .$$

1. Find the values of \(A\) and \(C\) and show that \(B = 0\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Simplify each term.

1. $$f ( x ) = ( 2 - 5 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 5 }$$ Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), as far as the term in \(x ^ { 3 }\), giving each coefficient as a simplified fraction.
(5)

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.

3. $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) } , \quad | x | < \frac { 2 } { 3 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$f ( x ) = \frac { A } { ( 3 x + 2 ) } + \frac { B } { ( 3 x + 2 ) ^ { 2 } } + \frac { C } { ( 1 - x ) }$$

1. find the values of \(B\) and \(C\) and show that \(A = 0\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). Simplify each term.
3. Find the percentage error made in using the series expansion in part (b) to estimate the value of \(\mathrm { f } ( 0.2 )\). Give your answer to 2 significant figures. \section*{LU}

1. (a) Find the binomial expansion of

$$\sqrt { } ( 1 - 8 x ) , \quad | x | < \frac { 1 } { 8 }$$ in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each term.
(b) Show that, when \(x = \frac { 1 } { 100 }\), the exact value of \(\sqrt { } ( 1 - 8 x )\) is \(\frac { \sqrt { } 23 } { 5 }\).
(c) Substitute \(x = \frac { 1 } { 100 }\) into the binomial expansion in part (a) and hence obtain an approximation to \(\sqrt { } 23\). Give your answer to 5 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. Given

$$f ( x ) = ( 2 + 3 x ) ^ { - 3 } , \quad | x | < \frac { 2 } { 3 }$$ find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.

1. (a) Find the binomial expansion of

$$\frac { 1 } { ( 4 + 3 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction. In the binomial expansion of $$\frac { 1 } { ( 4 - 9 x ) ^ { 3 } } , \quad | x | < \frac { 4 } { 9 }$$ the coefficient of \(x ^ { 2 }\) is \(A\).
(b) Using your answer to part (a), or otherwise, find the value of \(A\). Give your answer as a simplified fraction.

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.

2. $$f ( x ) = \frac { 3 x - 1 } { ( 1 - 2 x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ Given that, for \(x \neq \frac { 1 } { 2 } , \quad \frac { 3 x - 1 } { ( 1 - 2 x ) ^ { 2 } } = \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 1 - 2 x ) ^ { 2 } } , \quad\) where \(A\) and \(B\) are constants,

1. find the values of \(A\) and \(B\).
2. Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.
   (6)

1. $$f ( x ) = ( 3 + 2 x ) ^ { - 3 } , \quad | x | < \frac { 3 } { 2 }$$ Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), as far as the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction.
(5)

5. (a) Expand \(\frac { 1 } { \sqrt { } ( 4 - 3 x ) }\), where \(| x | < \frac { 4 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\). Simplify each term.
(b) Hence, or otherwise, find the first 3 terms in the expansion of \(\frac { x + 8 } { \sqrt { } ( 4 - 3 x ) }\) as a series in ascending powers of \(x\).

1. $$f ( x ) = \frac { 1 } { \sqrt { ( 4 + x ) } } , \quad | x | < 4$$ Find the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(6)

2. $$f ( x ) = \frac { 1 } { \sqrt { } \left( 9 + 4 x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { 2 }$$ Find the first three non-zero terms of the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\). Give each coefficient as a simplified fraction.

3. $$f ( x ) = \frac { 6 } { \sqrt { ( 9 - 4 x ) } } , \quad | x | < \frac { 9 } { 4 }$$

1. Find the binomial expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
   (6) Use your answer to part (a) to find the binomial expansion in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), of
2. \(\quad \mathrm { g } ( x ) = \frac { 6 } { \sqrt { } ( 9 + 4 x ) } , \quad | x | < \frac { 9 } { 4 }\)
3. \(\mathrm { h } ( x ) = \frac { 6 } { \sqrt { } ( 9 - 8 x ) } , \quad | x | < \frac { 9 } { 8 }\)

4. (a) Find the binomial expansion of $$\sqrt [ 3 ] { ( 8 - 9 x ) , \quad } \quad | x | < \frac { 8 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
(b) Use your expansion to estimate an approximate value for \(\sqrt [ 3 ] { 7100 }\), giving your answer to 4 decimal places. State the value of \(x\), which you use in your expansion, and show all your working.

1. (a) Find the binomial expansion of

$$\frac { 1 } { \sqrt { } ( 9 - 10 x ) } , \quad | x | < \frac { 9 } { 10 }$$ in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
Give each coefficient as a simplified fraction.
(b) Hence, or otherwise, find the expansion of $$\frac { 3 + x } { \sqrt { } ( 9 - 10 x ) } , \quad | x | < \frac { 9 } { 10 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
Give each coefficient as a simplified fraction.

1. Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
Give each coefficient as a fraction in its simplest form.
(6)