Consider $$(n-1)\exp\{(n-1)(\frac{1}{2}-p(n))\}(2p(n))^{\frac{n-1}{2}}$$
where $n$ is positive integer, $p(n)$ is a function of $n$ that is undefined. This expression in my context represents a probability of a random event, I want it to be 'small'.
My question is, under which condition of $p(n)$ is the above expression decreasingly (maybe exponentially decreasing) approaching $0$ when $n$ grows?
I have no experience of dealing with asymptotics, and here is what I tried: Assume $p(n)=n^\alpha$ (though I think this is too loose..) $$=(n-1)n^{\frac{\alpha(n-1)}{2}}e^{\frac{1}{2}(-2n^{\alpha+1}+2n^\alpha+n-1)}2^{\frac{n-1}{2}}\leq (n-1)n^{\frac{\alpha(n-1)}{2}}e^{\frac{1}{2}(-2n^{\alpha+1}+2n^\alpha+n-1)}e^{\frac{n-1}{2}}=(n-1)n^{\frac{\alpha(n-1)}{2}}e^{-n^{\alpha+1}+n^\alpha+n-1}$$
Many thanks for hints.