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How can I find a formula for $\sum_{i = 1}^{n} \frac{i}{K - i}$, $K \in \mathbb{Z}$, $K > n$?

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Writing $i/(K-i)$ as $K/(K-i)-1$, one sees that this sum is also $K(H_{K-1}-H_{K-n-1})-n, $ where $H_k=\displaystyle\sum_{i=1}^k\frac1i$ is the $k$th harmonic number. I doubt that any other general formula exists, and that this one has any practical use for fixed values of $K$ and $n$ (asymptotics being another matter).

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    There are potential approximations such as $K \log_e\left(1+\frac{n}{K-n-1}\right)-n$ and $\frac{n(n+1)}{2k}$.2011-07-10