Amazing Factorial Summation

Calculus Level 3

121!+2!+232!+3!+343!+4!+454!+5!+= ?\large\dfrac{1\cdot2}{1!+2!}+\dfrac{2\cdot3}{2!+3!}+\dfrac{3\cdot4}{3!+4!}+\dfrac{4\cdot5}{4!+5!}+\cdots=\ ?

Notation:
!! denotes the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

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