(a)[2]
Show, through calculation, that $\alpha$ lies between $0.5$ and $1$.
(b)[2]
Show that, should a sequence of positive values produced by the iterative formula $x_{n+1} = \frac{1}{3}\left(x_n + 4\tan^{-1}\left(\frac{1}{3x_n}\right)\right)$ converge, it converges to $\alpha$.
(c)[3]
Use this iterative formula to find $\alpha$ correct to $2$ decimal places. Record the outcome of each iteration to $4$ decimal places.