This is not what you think!

Algebra Level 4

$\large\dfrac{3}{1!+2!+3!} + \dfrac{4}{2!+3!+4!} +\cdots+ \dfrac{2016}{2014!+2015!+2016!}$

If the above expression can be represented as$\dfrac{1}{k!} - \dfrac{1}{(k+a)!} \; ,$ then find $$\dfrac{a}{k}$$.

Notation:

$$!$$ denotes the factorial notation. For example, $$8! = 1\times2\times3\times\cdots\times8$$.

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