I want to compare two following fractions $$ a = \frac{1}{411}\cdot\frac{1}{412}\cdot\frac{1}{413}; b = \frac{1}{63990006} $$
I want to compare these without any calculations, but simply manipulating, something like this:
$$ a=\frac{791}{993}; b=\frac{792}{991} $$
$$ a=\frac{791}{993}\cdot\frac{991}{991}=\frac{(792-1)\cdot(993-2)}{993\cdot991}=\frac{792\cdot993-2\cdot792-993+2}{993\cdot991}=\frac{792\cdot993-(2\cdot792+991)}{993\cdot993} $$
$$ b=\frac{792\cdot993}{991\cdot993} $$ so as the numerator of a is less than b's then $a < b$.
I am not sure if such an approach is applicable at all.
P.S. I see that the example provided does not need any manipulations, it's just for demonstration.
Note that
$$400^3=64\,000\,000$$
therefore
$$a = \frac{1}{411}\cdot\frac{1}{412}\cdot\frac{1}{413}< b = \frac{1}{63\,990\,006}$$