(a)[2]
By sketching an appropriate pair of graphs, demonstrate that the equation $\cosec\,\frac{1}{2}x = e^{x} - 3$ has exactly one root, called $\alpha$, in the interval $0 < x < \pi$.
(b)[2]
Confirm by calculation that $\alpha$ is between $1$ and $2$.
(c)[1]
Show that, if a sequence of values in the interval $0 < x < \pi$ defined by the iterative formula $x_{n+1} = \ln\left(\cosec\,\frac{1}{2}x_n + 3\right)$ converges, then its limit is $\alpha$.
(d)[3]
Apply the iterative formula with initial value $1.4$ to find $\alpha$ correct to $2$ decimal places. Record each iterative value to $4$ decimal places.
(e)[1]
State the least number of calculated iterations needed, using this initial value, to determine $\alpha$ correct to $2$ decimal places.