We study the reorientation transition in a thin ferromagnetic film bet
ween a perpendicularly magnetized state and an in-plane magnetized one
, at the growing of the temperature T and the number of planes N. In t
he framework of a renormalization group approach, we find the boundari
es between the three (perpendicularly magnetized, in-plane magnetized,
paramagnetic) regions. A maximum in the reorientation temperature T-R
(N) is predicted.