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Inspired by the recent measurement of the ratio of $B_c$ branching fractionsto $J/\psi \pi^+$ and $J/\psi \mu^+\nu_{\mu}$ final states at the LHCbdetector, we study the semileptonic decays of $B_c$ meson to the S-wave groundand radially excited 2S and 3S charmonium states with the perturbative QCDapproach. After evaluating the form factors for the transitions $B_c\rightarrowP,V$, where $P$ and $V$ denote pseudoscalar and vector S-wave charmonia,respectively, we calculate the branching ratios for all these semileptonicdecays. The theoretical uncertainty of hadronic input parameters are reduced byutilizing the light-cone wave function for $B_c$ meson. It is found that thepredicted branching ratios range from $10^{-6}$ up to $10^{-2}$ and could bemeasured by the future LHCb experiment. Our prediction for the ratio ofbranching fractions $\frac{\mathcal {BR}(B_c^+\rightarrow J/\Psi\pi^+)}{\mathcal {BR}(B_c^+\rightarrow J/\Psi \mu^+\nu_{\mu})}$ is in goodagreement with the data. For $B_c\rightarrow V l \nu_l$ decays, the relativecontributions of the longitudinal and transverse polarization are discussed indifferent momentum transfer squared regions. These predictions will be testedon the ongoing and forthcoming experiments.

Semileptonic decays of $B_c$ meson to S-wave charmonium states in the perturbative QCD approach