I already find the fact sum of reciprocal arithmetic progression with only positive term can't be integer. In the following $a $\sum_{0\le k\le n} \frac{1}{ak+d}\notin\mathbb{Z}$ But I'm interested in general case with negative term. Simple observation makes the constraint $\left|d\right|\ne 2$. $\sum_{-m\le k\le n} \frac{1}{ak+d}\notin\mathbb{Z}$?????? Is there any result about this?
sum of reciprocal arithmetic progression is not an integer with negative term.
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number-theory
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0Since $$\sum_{-m\le k\le n} \frac{1}{ak+d} = \sum_{0\le k\le n + m}\frac{1}{ak + (d-am)}$$ the answer will depend on how critical the condition that $a < d$ is. If the expression is also not integer when $a > d$, then it will not be integer for your expression either. If there are examples where $a > d$ and the sum is integer, then it is not guaranteed that your sum will also have examples, but it seems likely. – 2017-02-26