Sum of Binomial Coefficients over Lower Index/Corollary

Theorem

 * $\displaystyle \forall n \in \Z_{\ge 0}: \sum_{i \mathop \in \Z} \binom n i = 2^n$

where $\displaystyle \binom n i$ is a binomial coefficient.

Proof
From the definition of the binomial coefficient, when $i < 0$ and $i > n$ we have $\displaystyle \binom n i = 0$.

The result follows directly from Sum of Binomial Coefficients over Lower Index.