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Let A be a simple eigenvalue of a bounded linear operator T on a Banach space X, and let (T n ) be a resolvent operator approximation of T. For large n, let S n denote the reduced resolvent associated with T n and X n , the simple eigenvalue of T n near X. It is shown that sup under the assumption that all the spectral points of T which are nearest to X belong to the discrete spectrum of T. This is used to find error estimates for the Rayleigh-Schrodinger series for X and <p with initial termsdoi:10.1017/s1446788700030937 fatcat:jf3sgucdcjayljedhb6coyrrna