How does a perturbation theory make sense in quantum field theory?

The idea of a perturbation series in powers of a coupling $\alpha\ll1$ (for example, the fine structure constant in QED) make sense if the contribution of $(n+1)^{th}$ term in the series is smaller than the contribution of $n^{th}$ term. Now, the tree-level amplitude for a QED process is finite. However, at $\mathcal{O}(\alpha^2)$, the contribution from the S-matrix to the amplitude is divergent (all diagrams with loops are divergent because loop momentum can be arbitrarily high). How does perturbation theory make sense in QED (or in any quantum field theory)?