By the conclusion of # 4530, we have 

$$\frac{2}{(1-x)^3}=2 + (2\times 3)x + (3\times 4)x^2 + (4\times 5)x^3 + \times\qquad (|x| < 1)$$

Setting $x=-\frac{1}{2}$ to the above equality gives

$$\frac{2}{\left(1+\frac{1}{2}\right)^3}=2- \frac{2\times 3}{2} + \frac{3\times 4}{2^2} - \frac{4\times 5}{2^3} + \cdots$$

Dividing the above equation by $-2$ yields

$$-\frac{8}{27}=-\frac{1\times 2}{2}+ \frac{2\times 3}{2^2} - \frac{3\times 4}{2^3} + \frac{4\times 5}{2^4} + \cdots$$

Finally, adding $1$ to both sides leads to the final answer as 

$$1-\frac{8}{27}=\boxed{\frac{19}{27}}$$