I have to study alpha's value so that this integral
$$ \int_{0}^\infty{\frac{e^{-x}(1-\cos x)^\alpha}{x^{6\alpha +1}}dx} $$
converge. I was able to say that for $ x\to 0 $ it's $\alpha<0 $ but in the range of infinity I'm stuck, I don't know how to simplify the function because I can't do any asymptotic analysis or Taylor expansion. I thought that, because alpha is negative, $2\pi k$ is also a critical point and maybe I should make a Taylor expansion of $\cos x$ in the range of $2\pi$ and then use it when x is tending to infinity. I don't really know if it's the right thing, I hope someone will be able to help me, thank everyone.