The gradient corrected Perdew-Burke-Ernzerhof density functional in conjunction with a 3-21G basis set and periodic boundary conditions was employed to investigate the geometries and energies Of C(2)7 fluorinated armchair single wall carbon nanotubes (F-SWNT's) with diameters ranging from 16.4 to 4.2 Angstrom [(12,12) to (3,3)] as well as a C2F graphene sheet fluorinated on one side only. Using an isodesmic equation, we find that the thermodynamic stability of F-SWNT's increases with decreasing tube diameter. On the other hand, the mean bond dissociation energies of the C-F bonds increase as the tubes become thinner. The C-F bonds in the (5,5) F-SWNT's ate about as strong as those in graphite fluoride (CF), and are also covalent albeit slightly (<0.04 A) stretched. Whereas a fluorine atom is found not to bind covalently to the concave surface of [60]fullerene, endohedral covalent binding is possible inside a (5,5) SWNT despite a diameter similar to that of the C-60 cage.