# proof of binomial formula

Let $p\in\mathbb{R}$ and $x\in\mathbb{R},\;|x|<1$ be given. We wish to show that

 $(1+x)^{p}=\sum_{n=0}^{\infty}p^{\underline{n}}\;\frac{x^{n}}{n!},$

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

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

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

and note that

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

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

Title proof of binomial formula ProofOfBinomialFormula 2013-03-22 12:24:00 2013-03-22 12:24:00 rmilson (146) rmilson (146) 6 rmilson (146) Proof msc 26A06