Using the Euler-Maclaurin summation we calculate analytically the inte
rnal energy for non-interacting bosons confined within a harmonic osci
llator potential. The specific heat shows a sharp lambda-like peak ind
icating a condensation into the ground state at a well-defined transit
ion temperature. Full agreement is obtained with direct numerical calc
ulation of the same quantities. When the number of trapped particles i
s very large and at temperatures near and above the transition tempera
ture, the results also agree with previous approximate calculations. A
t extremely low temperatures both the specific heat and the number of
particles excited from the condensate are exponentially suppressed.