- (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 \(A\).