By sketching a suitable pair of graphs, show that the equation
$$\ln x = 2 - x ^ { 2 }$$
has only one root.
Verify by calculation that this root lies between \(x = 1.3\) and \(x = 1.4\).
Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)$$
converges, then it converges to the root of the equation in part (i).
Use the iterative formula \(x _ { n + 1 } = \sqrt { } \left( 2 - \ln x _ { n } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.