7.
$$f ( x ) = \frac { 8 - x } { ( 1 + x ) ( 2 - x ) } , \quad | x | < 1$$
- Express \(\mathrm { f } ( x )\) in partial fractions.
- Show that
$$\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x = \ln k$$
where \(k\) is an integer to be found.
- Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
7. continued
7. continued