Luminosity convergence of an outer surface-brightness profile ($I(R)\propto R^{-p}:L_{\mathrm{outer}}<\infty ⟺p>2$)

For a circular projected profile with a positive power-law tail, an annulus contributes $dF=2\pi I(R)RdR$. Thus the outer integral converges only when $p>2$; $p=2$ diverges logarithmically and $p<2$ as $R^{2-p}$. A shallower tail measured over a finite range must steepen or terminate before an infinite-radius extrapolation is physical. This projected criterion already includes line-of-sight integration and must not be replaced by a three-dimensional $r^{2}dr$ volume factor.