Reciprocal as Summation of Binomial Coefficients of Reciprocals

Theorem

 * $\forall n \in \Z_{>0}: \dfrac 1 n = \displaystyle \sum_{k \mathop = 0}^{n - 1} \left({-1}\right)^k \dbinom {n - 1} k \dfrac 1 {k - 1}$

where $\dbinom {n - 1} k$ denotes a binomial coefficient.

That is, for example: