Sum over k of n Choose k by x to the k by kth Harmonic Number/x = -1

Theorem
While for $x \in \R_{> 0}$ be a real number:


 * $\displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k = \left({x + 1}\right)^n \left({H_n - \ln \left({1 + \frac 1 x}\right)}\right) + \epsilon$

when $x = -1$ we have:


 * $\displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k = \dfrac {-1} n$

where:
 * $\dbinom n k$ denotes a binomial coefficient
 * $H_k$ denotes the $k$th harmonic number.

Proof
When $x = -1$ we have that $1 + \dfrac 1 x = 0$, so $\ln \left({1 + \dfrac 1 x}\right)$ is undefined.

Let $S_n = \displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k$

Then:

Setting $x = -1$: