proof of binomial formula

Let $p\in ℝ$ and be given. We wish to showthat

$${(1+x)}^p=\sum _{n=0}^{\mathrm{\infty }}{p}^{\underset{¯}{n}}\frac{x^n}{n!},$$

where $p^{\underset{¯}{n}}$ denotes the $n^{\text{th}}$ falling factorial of $p$.

The convergence of the series in the right-hand side of the aboveequation is a straight-forward consequence of the ratio test. Set

$$f(x)={(1+x)}^p.$$

and note that

$$f^{(n)}(x)=p^{\underset{¯}{n}}{(1+x)}^{p-n}.$$

The desired equality now follows from Taylor’s Theorem. Q.E.D.