I know that $\int\_0^{\infty}\sin(x^{\alpha})$ converges for $|\alpha| \geq 1$ which can be attributed to cancellation of areas and $\int\_0^{\infty}|\sin(x^{n})|$ diverges for $n = 2$ since there is no cancellation of areas. It seems unlikely but is there a $n$ such that the area of the oscillations can be made so small that it can be approximated by a geometric series and hence show that this integral converges?