  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.
