We investigate the bound states of relativistic positrons channeled in perfect single-walled carbon nanotubes (SWCNTs), with different chiral indices (n,m). The channeling potential, has been calculated according to the continuum model approximation given by Lindhard. We find a general expression for the channeling potential fit the calculated channeling potential with fitting parameters for each carbon nanotube. We calculate the energy eigenvalues by, using the WKB method, the energy eigenfunctions and the position probability density of the channeled positron in a plane normal to the nanotube axis by solving the Shrödinger equation using the obtained general formula for the channeling potential.