Convergence of cylindrical shock in argon is studied both experimentally and numerically. Shock tube experiments are conducted, where a planar shock is first transformed to a cylindrical shape and then converged to its focal axis. Numerical simulations of the converging shock using equations of state for an ideal gas and a real gas (SESAME 5173 model) are conducted and compared. High temporal resolution data of cylindrical shock convergence is presented. When comparing the trajectories of the converging shock of initial shock Mach number (M-S) of 4.63, the convergence exponent (alpha) in experiments is found to be 0.833. This alpha value in experiments is higher than the value obtained from computations with argon treated as an ideal gas but agrees well with the real gas computations. It is revealed that the form of convergence varies with different M-S. An asymptotic approach of alpha toward the self-similar solution for high M-S is attributed to an earlier transition of shock motion to self-similarity, while a significantly higher alpha observed at lower M-S is attributed to the negative influence of upstream nonuniformities and weaker initiation of the shock. It is found that even before the shock reflection, real gas effects are significant enough to affect the convergence of the shock and limit the extreme conditions predicted by the ideal gas computations. For an M-S of 4.63, the maximum temperature reached is 9250 K before reflection, leading to 0.12% of the argon gas undergoing the first stage of ionization.