Probability limit theory is one of the branches of probability and also is important basic theory of the science of probability and statistics. Limit theory mainly study independent random variables, but in many practical problems, samples are not independent, or the function of independent sample is not independent, or the verification of independent is more difficult. So the concept of dependent random variables in probability and statistics is mentioned. Exchangeable random variables are a major type of dependent random variable. In this paper we extend the Baum and Katz theorem in the condition of independent, and obtain the specific forms of expression of Baum and Katz theorem in the case of exchangeable random variables.