An Infinite Partition for 2016

The set of non-negative integers are partitioned into arithmetic sequence with first terms \(a_1 , a_2, \ldots, a_k\), with common differences, \(d_1, d_2,\ldots, d_k\), respectively. If \(k = 2016\), find \[\frac{a_1}{d_1} + \frac{a_2}{d_2} +\cdots + \frac{a_k}{d_k}.\]

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