I obtained the following formula in Mathematica:
$$\frac{d}{dn}\ln\binom{n}{k} = H_{n} – H_{n-k}$$
where $H_n$ are the harmonic numbers ($H_n = \sum_{i=1}^n 1/i$). But I have no idea how to prove it. Can someone help me? Or at least provide a reference to a textbook/paper?
Best Answer
\begin{eqnarray*} \binom{n}{k} &=& \frac{n(n-1) \cdots(n-k+1)}{k!} \\ \ln \binom{n}{k} &=&\ln n + \ln(n-1) + \cdots +\ln(n-k+1) -\ln(k!) \\ \frac{d}{dn} \ln \binom{n}{k} &=& \frac{1}{n} + \frac{1}{n-1} +\cdots + \frac{1}{n-k+1} =\color{red}{H_n-H_{n-k}}. \end{eqnarray*}