Deal With These Sums Separately

Algebra Level 3

\[\sum_{j=2}^{2016} \sum_{k=1}^{j-1} \dfrac kj = \frac{1}{2}+\frac{1}{3}+\frac{2}{3}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\cdots+\frac{2013}{2016}+\frac{2014}{2016}+\frac{2015}{2016}= \, ? \]

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