Binomial Theorem/General Binomial Theorem/Proof 1

Proof
Let $R$ be the radius of convergence of the power series:
 * $\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n$

Then:

Thus for $\size x < 1$, Power Series is Differentiable on Interval of Convergence applies:


 * $\ds D_x \map f x = \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} n x^{n - 1}$

This leads to:

Gathering up:
 * $\paren {1 + x} D_x \map f x = \alpha \map f x$

Thus:
 * $\map {D_x} {\dfrac {\map f x} {\paren {1 + x}^\alpha} } = -\alpha \paren {1 + x}^{-\alpha - 1} \map f x + \paren {1 + x}^{-\alpha} D_x \map f x = 0$

So $\map f x = c \paren {1 + x}^\alpha$ when $\size x < 1$ for some constant $c$.

But $\map f 0 = 1$ and hence $c = 1$.