Sum over k of n Choose k by x to the k by kth Harmonic Number

Theorem
Let $x \in \R_{> 0}$ be a real number.

Then:


 * $\ds \sum_{k \mathop \in \Z} \binom n k x^k H_k = \paren {x + 1}^n \paren {H_n - \map \ln {1 + \frac 1 x} } + \epsilon$

where:
 * $\dbinom n k$ denotes a binomial coefficient
 * $H_k$ denotes the $k$th harmonic number
 * $0 < \epsilon < \dfrac 1 {x \paren {n + 1} }$

Proof
Let $S_n := \ds \sum_{k \mathop \in \Z} \binom n k x^k H_k$.

Then:

As $S_1 = x$, we have that:

From Corollary to Power Series Expansion for $\map \ln {1 + x}$ we have:
 * $\ds \map \ln {1 - z} = -\sum_{k \mathop \ge 1} \frac {z^k} k$

Hence by Logarithm of Reciprocal we have:
 * $\ds \ln \frac 1 {1 - z} = -\map \ln {1 - z} = \sum_{k \mathop \ge 1} \frac {z^k} k$

Thus:

So as $n \to \infty$, $\ds \sum_{k \mathop \ge 1} \frac 1 {k \paren {x + 1}^k}$ converges, and so:

The result follows.