Show that \(\frac { x } { ( 1 - x ) ^ { 3 } } \approx x + 3 x ^ { 2 } + 6 x ^ { 3 }\) for small values of \(x\).
Use this result, together with a suitable value of \(x\), to obtain a decimal estimate of the value of \(\frac { 100 } { 729 }\).
Show that \(\frac { x } { ( 1 - x ) ^ { 3 } } = - \frac { 1 } { x ^ { 2 } } \left( 1 - \frac { 1 } { x } \right) ^ { - 3 }\). Hence find the first three terms of the binomial expansion of \(\frac { x } { ( 1 - x ) ^ { 3 } }\) in powers of \(\frac { 1 } { x }\).
Comment on the suitability of substituting the same value of \(x\) as used in part (ii) in the expansion in part (iii) to estimate the value of \(\frac { 100 } { 729 }\).