Series expansion of rational function

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)

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. $$f ( x ) = ( 3 + 2 x ) ^ { - 3 } , \quad | x | < \frac { 3 } { 2 }$$ 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)

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)

1. Use the binomial series to find the expansion of

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

1 Find the coefficient of the term in \(x ^ { 3 }\) in the expansion of \(\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }\).

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.

2．（a）For the binomial expansion of \(\frac { 1 } { ( 1 - x ) ^ { 2 } } , | x | < 1\) ，in ascending powers of \(x\) ，
（i）find the first four terms，
（ii）write down the coefficient of \(x ^ { n }\) ．
（b）Hence，show that，for \(| x | < 1 , \sum _ { n = 1 } ^ { \infty } n x ^ { n } = \frac { x } { ( 1 - x ) ^ { 2 } }\) ．
（c）Prove that，for \(| x | < 1 , \sum _ { n = 1 } ^ { \infty } ( a n + 1 ) x ^ { n } = \frac { ( a + 1 ) x - x ^ { 2 } } { ( 1 - x ) ^ { 2 } }\) ，where \(a\) is a constant．
（d）Hence evaluate \(\sum _ { n = 1 } ^ { \infty } \frac { 5 n + 1 } { 2 ^ { 3 n } }\) ．

3 In this question you should assume that \(- 1 < x < 1\).

1. For the binomial expansion of \(( 1 - x ) ^ { - 2 }\)
   1. find and simplify the first four terms,
   2. write down the term in \(x ^ { n }\).
2. Write down the sum to infinity of the series \(1 + x + x ^ { 2 } + x ^ { 3 } + \ldots\).
3. Hence or otherwise find and simplify an expression for \(2 + 3 x + 4 x ^ { 2 } + 5 x ^ { 3 } + \ldots\) in the form \(\frac { a - x } { ( b - x ) ^ { 2 } }\) where \(a\) and \(b\) are constants to be determined.