Let $c>0$ and $r >0$ and consider the integral for $n \geq 2$
$$ I_{r,n} (c) =\int_{1}^{\infty} \frac{\exp\left( - (y-c)^n\right)}{y^r} \mathrm{d} y$$
How do I show $I_{r,n} (c) < \infty$?
I am not sure if this is even true without making additional assumptions on $r,n$ or $c$