Consider the function
$(1-q)(1-q)^{2n-2}(1-q^2)^{(r-2)n}$
where $n$ is a (large) positive integer, $r$ is a (small) positive integer, and $q$ is a real number in [0,1] (is a probability). I have the feeling that this function should be very close to 0 after $q$ reaches some threshold, say $1/n$. Is there any way to prove my intuition right?
I know that the function converges to 0.