Difficult math question

$\sum \limits_{n=0}^\infty \dfrac{n!}{(n+1+\delta)!} = \dfrac{1}{\delta \cdot \delta!}\\ 2\sum \limits_{n=0}^\infty \dfrac{n!}{(n+3)!} = \dfrac{2}{2\cdot 2!} = \dfrac 1 2$

Compute the sum

$\frac{2}{1 \cdot 2 \cdot 3} + \frac{2}{2 \cdot 3 \cdot 4} + \frac{2}{3 \cdot 4 \cdot 5} + \cdots$

answer see: https://web2.0calc.com/questions/algebra_47009#r4