We use density functional computations to study the size effects on the structural, electronic, magnetic, and optical properties of (5,0) finite carbon nanotubes (FCNT), with length in the range of 4-44 Å. It is found that the structural and electronic properties of (5,0) FCNTs, in the ground state, converge at a length of about 30 Å, while the excited state properties exhibit long-range edge effects. We discuss that curvature effects govern the electronic structure of short (5,0) FCNTs and enhance energy gap of these systems, in contrast to the known trend in the periodic limit. It is seen that compensation of curvature effects in two special small sizes, may give rise to spontaneous magnetization. The obtained cohesive energies provide some insights into the effects of environment on the growth of FCNTs. The second-order difference of the total energies reveals an important magic size of about 15 Å. The optical and dynamical magnetic responses of the FCNTs to polarized electromagnetic pulses are studied by time dependent density functional theory. The obtained results show that the static and dynamic magnetic properties mainly come from the edge carbon atoms. The optical absorption properties are described in term of local field effects and characterized by Casida linear response calculations.