Amazing Factorial Summation

Calculus Level 3

\[\large\dfrac{1\cdot2}{1!+2!}+\dfrac{2\cdot3}{2!+3!}+\dfrac{3\cdot4}{3!+4!}+\dfrac{4\cdot5}{4!+5!}+\cdots=\ ?\]

Notation:
\(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).