This paper proposes a novel finite memory estimation based learning algorithm (FME-LA) for recurrent neural networks (RNNs) to accurately identify unknown nonlinear systems. The proposed algorithm, FME-LA, is designed through finite memory estimation (FME) whose gain is obtained under the unbiased condition by minimizing the Frobenius norm. The FME is designed on the concept of the horizon to decide how many recent measurements are considered and maintain a finite memory structure of the learning algorithm. Therefore, the proposed algorithm provides accurate performance and fast convergence for the system identification of unknown nonlinear systems under rapidly or smoothly changing circumstances with inaccurate information, such as modeling uncertainties or incorrect noise statistics. We confirm fast convergence and accurate performance of the proposed algorithm through simulation and experimental results.