4.
$$\mathrm { f } ( x ) = ( 1 + 3 x ) ^ { - 1 } , | x | < \frac { 1 } { 3 }$$
- Expand \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
- Hence show that, for small \(x\),
$$\frac { 1 + x } { 1 + 3 x } \approx 1 - 2 x + 6 x ^ { 2 } - 18 x ^ { 3 }$$
- Taking a suitable value for \(x\), which should be stated, use the series expansion in part (b) to find an approximate value for \(\frac { 101 } { 103 }\), giving your answer to 5 decimal places.