What kind of geometric series is $f(m) = \sum_{r=1}^{m}\frac{r+1}{n-r-1}$?

Question

I have a geometric series that looks like $$ f(m) = \sum_{r=1}^{m}\frac{r+1}{n-r-1} \\ = \frac{2}{n-2} + \frac{3}{n-3} + \frac{4}{n-4} + \cdots + \frac{m+1}{n-(m-1)} \\ $$ I wrote out a few terms to see if I can identify a simplification but I couldn't. What kind of geometries series is this, and how can I simplify to get a closed form expression

Answer 1

I'm assuming $n$ is fixed. Note that in order for the sum to be well-defined, we must have $n \leq m$.

A closed form is unlikely, but: \begin{align*} f(m) &= \sum_{r=1}^m \frac{r+1}{n - (r + 1)} \\ &= \sum_{r=1}^m \left(\frac{n}{n - (r + 1) }- 1\right) \\ &= n\sum_{r=1}^m \frac{1}{n - (r + 1)} - m \\ &= n(H_{n-2} - H_{n-m-2}) - m \end{align*} where $H_k$ is the $k^\text{th}$ harmonic number.