It is given that the equation \(\mathrm { e } ^ { 2 x } = 5 + \cos 3 x\) has only one root.
Show by calculation that this root lies in the interval \(0.7 < x < 0.8\).
Show that if a sequence of values in the interval \(0.7 < x < 0.8\) given by the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 5 + \cos 3 x _ { n } \right)$$
converges then it converges to the root of the equation in part (a).
Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.