Tallying Tiny Logs

$\begin{aligned} S & = \sum_{n=2}^{2016} \frac {2016n}{\log_n (2016!)^n} = \sum_{n=2}^{2016} \frac {2016n}{\frac {n \log (2016!)}{\log n}} = \sum_{n=2}^{2016} \frac {2016 \log n}{\log (2016!)} =\frac {2016 \log (2016!)}{\log (2016!)} = \boxed{2016} \end{aligned}$

$S=\dfrac{2\times2016}{\log_{2}(2016!)^{2}}+\dfrac{3\times2016}{\log_{3}(2016!)^{3}}+\dfrac{4\times2016}{\log_{4}(2016!)^{4}}+\dots+\dfrac{2016\times2016}{\log_{2016}(2016!)^{2016}}$

$S=2016(\dfrac{2}{2\log_{2}2016!}+\dfrac{3}{3\log_{3}2016!}+\dfrac{4}{4\log_{4}2016!}+\dots+\dfrac{2016}{2016\log_{2016}2016!})$

$S=2016(\dfrac{1}{\log_{2}2016!}+\dfrac{1}{\log_{3}2016!}+\dfrac{1}{\log_{4}2016!}+\dots+\dfrac{1}{\log_{2016}2016!})$

$S=2016(\dfrac{\log2}{\log2016!}+\dfrac{\log3}{\log2016!}+\dfrac{\log4}{\log2016!}+\dots+\dfrac{\log2016}{\log2016!})$

$S=2016(\dfrac{\log2+\log3+\log4+\dots+\log2016}{\log2016!})$

$S=2016(\dfrac{\log(2\times3\times4\times\dots\times2016)}{\log2016!})$

$S=2016(\dfrac{\log2016!}{\log2016!})$

$S=2016(1)$

$S=2016$

$S=\displaystyle \sum_{n=2}^{2016} \frac{2016n}{log_{n}(2016!)^n} = \sum_{n=2}^{2016} \frac{2016n \cdot log_{2016!}(n)}{n} =2016\cdot log_{2016!}(2016!)=2016$