Spectrum of density perturbations produced during inflation to leading order in a general slow-roll approximation

The standard calculation of the spectrum of density perturbations produced during inflation assumes (i) $|n-1|\ll 1$ and (ii) $|dn/d\ln k|\ll |n-1|.$ Slow-roll predicts and observations require (i), but neither slow-roll nor observations require (ii). In this paper I derive formulas for the spectrum $\mathcal{P},$ spectral index $n,$ and running of the spectral index $dn/d\ln k,$ assuming only (i) and not (ii). I give a large class of observationally viable examples in which these general slow-roll formulas are accurate but the standard slow-roll formulas for the spectral index and running are incorrect.