8 It is required to solve the equation \(\ln ( x - 1 ) - x + 3 = 0\).
You are given that there are two roots, \(\alpha\) and \(\beta\), where \(1.1 < \alpha < 1.2\) and \(4.1 < \beta < 4.2\).
- The root \(\beta\) can be found using the iterative formula
$$x _ { n + 1 } = \ln \left( x _ { n } - 1 \right) + 3$$
(a) Using this iterative formula with \(x _ { 1 } = 4.15\), find \(\beta\) correct to 3 decimal places. Show all your working.
(b) Explain with the aid of a sketch why this iterative formula will not converge to \(\alpha\) whatever initial value is taken. - (a) Show that the Newton-Raphson iterative formula for this equation can be written in the form
$$x _ { n + 1 } = \frac { 3 - 2 x _ { n } - \left( x _ { n } - 1 \right) \ln \left( x _ { n } - 1 \right) } { 2 - x _ { n } }$$
(b) Use this formula with \(x _ { 1 } = 1.2\) to find \(\alpha\) correct to 3 decimal places.