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Sums of fading envelopes occur in several wireless communications applications, such as equal-gain combining, signal detection, outage probability, intersymbol interference, etc. The exact evaluation of the sum statistics is known to be very intricate. One of the purposes of this Letter is to provide highly accurate closed-form approximations to the probability density function and cumulative distribution function of the sum of independent identically distributed (i.i.d.) alpha-mu (generalized gamma) variates. Based on and as an extension of such an approach, simple precise approximations for the performance metrics of equal-gain combining and maximal-ratio combining receivers operating on i.i.d. alpha-mu fading channels are proposed. Samples examples are given to illustrate that, for practical purposes, exact and approximate solutions are indistinguishable from each other.