Establish the binomial series in (12) by means of the following steps.

(a) Show that y = (1 + x)^α satisfies the initial value problem (1 + x)y' = αy, y(0) = 1.

(b) Show that the power series method gives the binomial series in (12) as the solution of the initial value problem in part (a), and that this series converges if |x| < 1.

(c) Explain why the validity of the binomial series given in (12) follows from parts (a) and (b).