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Sums of Rayleigh random variables occur extensively in wireless communications. A closed-form expression does not exist for the sum distribution and consequently, it is often evaluated numerically or approximated. A widely used small argument approximation for the density is shown to be inaccurate for medium and large values of the argument. Highly accurate, simple closed-form expressions for the sum distributions and densities are presented. These approximations are valid for a wide range of argument values and number of summands.