By sketching a suitable pair of graphs, show that there is only one value of \(x\) that is a root of the equation
$$\frac { 1 } { x } = \ln x$$
Verify by calculation that this root lies between 1 and 2 .
Show that this root also satisfies the equation
$$x = \mathrm { e } ^ { \frac { 1 } { x } }$$
Use the iterative formula
$$x _ { n + 1 } = \mathrm { e } ^ { \frac { 1 } { x _ { n } } }$$
with initial value \(x _ { 1 } = 1.8\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.