The Rayleigh-Taylor instability (RTI) in cylindrical geometry is investigated analytically through a second-order weakly nonlinear (WN) theory considering the Bell-Plesset (BP) effect.The goveing equations for the combined perturbation growth are derived.The WN solutions for an exponentially convergent cylinder are obtained.It is found that the BP and RTI growths are strongly coupled,which results in the bubble-spike asymmetric structure in the WN stage.The large Atwood number leads to the large deformation of the convergent interface.The amplitude of the spike grows faster than that of the bubble especially for large mode number m and large Atwood number A.The averaged interface radius is small for large mode number perturbation due to the mode-coupling effect.