A standard perturbation result states that perturbed eigenvalues and eigenprojections admit a perturbation series provided that the operator norm of the perturbation is smaller than a constant times the corresponding eigenvalue isolation distance. In this paper, we show that the same holds true under a weighted condition, where the perturbation is symmetrically normalized by the square-root of the reduced resolvent. This weighted condition originates in random perturbations where it leads to significant improvements.