4 The equation \(x ^ { 3 } - x - 3 = 0\) has one real root, \(\alpha\).
- Show that \(\alpha\) lies between 1 and 2 .
Two iterative formulae derived from this equation are as follows:
$$\begin{aligned}
& x _ { n + 1 } = x _ { n } ^ { 3 } - 3
& x _ { n + 1 } = \left( x _ { n } + 3 \right) ^ { \frac { 1 } { 3 } }
\end{aligned}$$
Each formula is used with initial value \(x _ { 1 } = 1.5\). - Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.