Summing Factorials

Calculus Level 4

The value of

11!(1+1!)(1+2!)+22!(1+2!)(1+3!)+33!(1+3!)(1+4!)+20142014!(1+2014!)(1+2015!)\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 A!A!+1BC,\dfrac{A!}{A!+1}-\dfrac{B}{C}, where A,BA,B and C C are coprime positive integers, and A>B,C.A>B,C. Find the value of A+B2C.\dfrac{A+B}{2C}.

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