Increasing Alternating Sum of Binomial Coefficients

Theorem
Let $n \in \Z$ be an integer.

Then:
 * $\displaystyle \sum_{j \mathop = 0}^n \left({-1}\right)^{n+1} j \binom n j = 0$

where $\displaystyle \binom n k$ denotes a binomial coefficient.

That is:
 * $\displaystyle 1 \binom n 1 - 2 \binom n 2 + 3 \binom n 3 - \cdots + \left({-1}\right)^{n+1} n \binom n n = 0$