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A train of periodic waves propagating on a non-uniform current which possesses an exponential distribution of vorticity is investigated by a relatively new analytic method for general nonlinear problems, namely the homotopy analysis method (HAM).Convergent series solutions are derived for not only small amplitude waves on a weak current but also large amplitude waves on a relatively strong current.The dispersion relationships in cases of different currents are compared each other.It is found that, for a co-flowing current, the vorticity tends to increase the wave phase speed and decrease the wave height.For an opposite current, the vorticity tends to decrease the wave phase speed but increase the wave height.This work also shows the potential of the homotopy analysis method for complicated problems with strong nonlinearity.