The non-extensive statistical mechanics has been used to describe a variety of complex systems. The maximization of entropy, often used to introduce the non-extensive statistical mechanics, is a formal procedure and does not easily leads to physical insight. In this article we investigate the canonical ensemble in the non-extensive statistical mechanics by considering a small system interacting with a large reservoir via short-range forces and assuming equal probabilities for all available microstates. We concentrate on the situation when the reservoir is characterized by generalized entropy with non-extensivity parameter q>1. We also investigate the problem of divergence in the non-extensive statistical mechanics occurring when q>1 and show that there is a limit on the growth of the number of microstates of the system that is given by the same expression for all values of q.
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