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How to estimate the following sum in terms of $n$?

$ \sum_{j_1,\ldots, j_{2k}\neq n}\frac{1}{(n-j_1)(n-j_2)\cdots(n-j_{2k-1})(n-j_{2k})}$ with $n+j_1, j_1-j_2, \ldots, j_{2k}-j_{2k-1}, n-j_{2k} \in \{-2,2\}$.

Do you have any ideas and how to proceed?

Thanks.

  • 0
    Yes, it converges to $0$ but how fast? Can we express in terms of $n$?2012-12-04

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You donot say the lower bound of j and I suppose $j \ge 0$ and proved this: $\sum_{0\le j\le n}\frac{1}{|n-j|}=1+(H_n-1)\le 1+\int_1^n \frac{1}{x} dx=1+\log n$ and it is not difficult to prove the statement then.