Numerical estimation

Question

Show that the numerical value of the expression

$\frac{2\times4\times6\times\dots\times2020}{1\times3\times5\times\dots\times2019}$

is between 44 and 64.

Answer 1

Note that $$\frac{2\cdot4\cdot\ldots\cdot(2n)}{1\cdot3\cdot\ldots\cdot(2n-1)} =\frac{2^{2n}(n!)^2}{(2n)!}$$

Then use the Stirling formula ($n=1010$) $$ n! \sim \sqrt {2\pi n} n^n e^{-n} $$