Summing Factorials

Calculus Level 4

The value of

\[\dfrac{1 \cdot 1!}{(1+1!)(1+2!)}+\dfrac{2 \cdot 2!}{(1+2!)(1+3!)}+\dfrac{3 \cdot 3!}{(1+3!)(1+4!)}+ \cdots\frac{2014\cdot2014!}{(1+2014!)(1+2015!)}\]

can be expressed in the form \(\dfrac{A!}{A!+1}-\dfrac{B}{C},\) where \(A,B\) and \( C\) are coprime positive integers, and \(A>B,C.\) Find the value of \(\dfrac{A+B}{2C}.\)

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