An approximate analytical method for the evaluation of the cumulative distribution function (CDF) of the sum of L independent random variables (RVs) is presented. The proposed method is based on the convergent infinite series approach، which makes it possible to describe the CDF in the form of an infinite series. The computation of the coefficients of this series needs complicated integrations over the RV’s probability density function (PDF). In some cases، the required integrations have closed-form in terms of confluent hypergeometric function and in other cases، the required integrations can not be analytically solved and have not a closed-form solution. In this paper، an approximation method for computation of the coefficients of the CDF series is presented that only needs the mean and the variance of the RV، so it has low computational complexity; it eliminates the need for calculation of complex functions and can be used as a unified tool for determining CDF of a sum of statistically independent RVs. To present an application for the developed approximation method، it is used to find the distribution of the sum of generalized Gamma (GG) RVs. The derived approximate expressions are used in the performance analysis of equal-gain combining (EGC) receivers operating over GG fading channels. The accuracy of the developed approximation method is verified by performing comparisons between exact existing results in the literature and computer simulations results.