Increasing Alternating Sum of Binomial Coefficients

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

Then:
 * $\ds \sum_{j \mathop = 0}^n \paren {-1}^{n + 1} j \binom n j = 0$

where $\dbinom n k$ denotes a binomial coefficient.

That is:
 * $1 \dbinom n 1 - 2 \dbinom n 2 + 3 \dbinom n 3 - \cdots + \paren {-1}^{n + 1} n \dbinom n n = 0$