8 The iteration \(x _ { n + 1 } = \frac { 1 } { \left( x _ { n } + 2 \right) ^ { 2 } }\), with \(x _ { 1 } = 0.3\), is to be used to find the real root, \(\alpha\), of the equation \(x ( x + 2 ) ^ { 2 } = 1\).
- Find the value of \(\alpha\), correct to 4 decimal places. You should show the result of each step of the iteration process.
- Given that \(\mathrm { f } ( x ) = \frac { 1 } { ( x + 2 ) ^ { 2 } }\), show that \(\mathrm { f } ^ { \prime } ( \alpha ) \neq 0\).
- The difference, \(\delta _ { r }\), between successive approximations is given by \(\delta _ { r } = x _ { r + 1 } - x _ { r }\). Find \(\delta _ { 3 }\).
- Given that \(\delta _ { r + 1 } \approx \mathrm { f } ^ { \prime } ( \alpha ) \delta _ { r }\), find an estimate for \(\delta _ { 10 }\).