The shock-bubble interaction is a basic configuration for studying the more general case of shock-accelerated inhomogeneous flows. In previous studies, a planar shock wave interacting with a spherical gas bubble was extensively investigated, in which the effects of shock intensity, Atwood number and secondary shock on the bubble development were considered and elucidated. However, in most of practical applications, such as inertial confinement fusion, a converging shock wave is generally involved. It is therefore of fundamental interest to explore the perturbation growth under converging shock conditions. Due to the difficulties encountered in generating a perfectly converging shock wave in laboratory, experimental investigation on the converging shock-accelerated inhomogeneous flows was seldom carried out previously. The preliminary study on the development of a gas bubble impacted by a converging shock wave showed that a large discrepancy exists compared with the planar counterparts. Because of the intrinsic three-dimensional (3D) features of this problem, the current experimental techniques are inadequate to explore the detailed differences between planar and converging shocks accelerating gas bubbles. As a result, numerical simulations become important and necessary. In this work, evolution of an SF6 spherical gas bubble surrounded by air accelerated by a cylindrical converging shock wave and a planar shock wave is numerically investigated by a 3D program, focusing on the convergence effect on the interface evolution. Multi-component compressible Euler equations are adopted in the 3D program and the finite volume method is used. The MUSCL-Hancock scheme, a second-order upwind scheme, is adopted to achieve the second-order accuracy on both temporal and spatial scales. Compared with planar shock wave, a cylindrical converging shock wave has curvature, and as the converging shock wave moves forward, the shock strength and the wall effect both increase, which will result in the diversity of the flow field after shock impact. The numerical results show that the vortex rings formed under converging shock condition are sharper than those under planar shock condition which may be associated with geometric contraction effect of the tube and reflected shock from the wall. Besides, the peak pressure generated in the vicinity of the downstream pole of the bubble under converging shock condition is higher than that of planar shock wave, and, therefore, the jet induced by high pressures moves faster under converging shock condition. Due to the variations of shock curvature and shock intensity, the distribution law and amplitude of vorticity generated by converging shock wave at the interface is changed. Comparison between circulation and gas mixing rate indicates that the converging shock is beneficial to promoting vorticity generation and gas mixing. From the present work, it can be concluded that the convergence effect plays an important role in interface evolution.