  It is well known that at the thermodynamic limit there are no observable
differences in the results obtained by grand canonical and canonical
descriptions of a many-body system. In the present paper, we test the validity
of this statement for finite systems using as an example an ensemble of bosons
trapped in a 1D harmonic potential well. We have found an analytical formula
for the canonical partition function and shown that, for 100 trapped atoms, the
discrepancy between the grand canonical and canonical predictions for the
condensate fraction reaches 10% in the vicinity of the Bose-Einstein threshold.
This discrepancy decreases only logarithmically as the number of atoms
increases. Furthermore we investigate numerically the case of a 3D
"cigar-shape" trap in the range of parameters corresponding to current BEC
experiments.
