Sum over k to n of k Choose m by kth Harmonic Number

Theorem

 * $\displaystyle \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \left({H_{n + 1} - \frac 1 {m + 1} }\right)$

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

Proof
First we note that by Pascal's Rule:


 * $\dbinom k m = \dbinom {k + 1} {m + 1} - \dbinom k {m + 1}$

Thus: