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This paper analyzes the performance of an amplify-and-forward (AF) protocol based cooperative multi-hop relay networks (CMHNs), where signals from the multi-hop relay network (MHN) and the direct path are considered in the detection, over the flat Nakagami-m fading channels. The analysis is accomplished by using the Beaulieu series that obtains a cumulative distributed function (CDF) through a characteristic function (CHF). Since the received SNR of MHN is mainly affected by the weakest channel in the network, we propose a hop selection scheme that prevents the performance degradation of the MHNs. We expand the CDF analysis to the networks employing proposed scheme by a tight bound of the received SNR. Also, in order to evaluate the diversity and coding gains of the proposed hop selection scheme, an asymptotic outage probability at high SNR region is derived. We show that the simulation results are closely matched to the analyzed results.