By sketching a suitable pair of graphs on the same diagram, show that the equation
$$\ln x = 2 \mathrm { e } ^ { - x }$$
has exactly one root.
Verify by calculation that the root lies between 1.5 and 1.6.
Show that if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \mathrm { e } ^ { 2 \mathrm { e } ^ { - x _ { n } } }$$
converges, then it converges to the root of the equation in part (a).
Use the iterative formula in part (c) to determine the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.