What is the value of the following sums/differences of infinite ratio of products? $$1-\frac{3}{4}+\frac{3\times5}{4\times8}-\frac{3\times5\times7}{4\times8\times12}+\frac{3\times5\times7\times9}{4\times8\times\times12\times16}-\dotsb $$
Seriously, I have no idea how to approach this problem. Any help is welcome.

We have from the general binomial expansion: $$\frac1{(1+x)^r} = 1-r x+\frac{r(r+1)}{1\cdot 2}x^2 - \frac{r(r+1)(r+2)}{1\cdot 2 \cdot 3}x^3 + \cdots$$
Comparing terms, we see $x = \frac12, r = \frac32$ works.
