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The differential evolution algorithm (DE) is one of the most powerful stochastic real-parameter optimization algorithms. The theoretical studies on DE have gradually attracted the attention of more and more researchers. However, few theoretical researches have been done to deal with the convergence conditions for DE. In this paper, a sufficient condition and a corollary for the convergence of DE to the global optima are derived by using the infinite product. A DE algorithm framework satisfying the convergence conditions is then established. It is also proved that the two common mutation operators satisfy the algorithm framework. Numerical experiments are conducted on two parts. One aims to visualize the process that five convergent DE based on the classical DE algorithms escape from a local optimal set on two low dimensional functions. The other tests the performance of a modified DE algorithm inspired of the convergent algorithm framework on the benchmarks of the CEC2005. [ABSTRACT FROM AUTHOR]