Approximate $\sum\limits_{k=0}^{m-1}\frac{k}{m-k}$
How do I approximate (or better find a closed formed formula) for $\sum\limits_{k=0}^{m-1}\frac{k}{m-k}$, where $m$ is large.
This looks suspiciously like a partial sum for the harmonic series, but I do know how to find a closed form for it. (Unless I missed something obvious).
Any help or insights is appreciated
$ \textbf{Hint :} $ \begin{aligned}\sum_{k=0}^{m-1}{\frac{k}{m-k}}&=\sum_{k=1}^{m}{\frac{m-k}{k}}\\ &=mH_{m}-m\end{aligned}
The closed-form solution is $$\gamma m-m+m \psi ^{(0)}(m)+1$$
using the Euler gamma and polygamma functions:
