Classical computations can not capture the essence of infinite computations very well. This paper will focus on a class of infinite computations called convergent infinite computations and establish a logical framework for describing and analyzing how an infinite computation interacts and evolves in changing environments and what the limit of the evolution might be. A logic for convergent infinite computations was proposed by extending first order theories using Cauchy sequences,which has stronger expression power than the first order logic. A computation model,called procedure scheme,for convergent infinite computations was proposed,on the basis of classical Turing machine and formal theory sequences and their limits. It has stronger computing power than Turing machines and real machines in the sense of limit computations. As an example of application of the above study,the limit behavior of data mining was discussed by means of limits of theory sequences.