The method of effective-potential expansion is employed to describe superconductivity in strongly correlated electron systems. We have obtained a gap equation in which both the pairing potential ${V}_{p}$(k,k') and the single-particle energy \ensuremath{\varepsilon}${\ifmmode \tilde{}\else \~{}\fi{}}_{\mathrm{k}}$ are expanded in terms of the effective potential. When the ring approximation or the random-phase approximation is used to evaluate each term in the expansion, ${V}_{p}$(k,k') and \ensuremath{\varepsilon}${\ifmmode \tilde{}\else \~{}\fi{}}_{\mathrm{k}}$ have the form ${V}_{p}^{\mathrm{KMK}(\mathrm{k}}$,k')/${z}_{\mathrm{k}}$${z}_{\mathrm{k}\mathcal{'}}$, and ${\ensuremath{\varepsilon}}_{\mathrm{k}}^{\mathrm{*}}$/${z}_{\mathrm{k}}$, respectively, where ${V}_{p}^{\mathrm{KMK}}$ is the pairing potential derived by Kirzhnits, Maksimov, and Khomskii (KMK), ${z}_{\mathrm{k}}$ is the renormalization factor, and ${\ensuremath{\varepsilon}}_{\mathrm{k}}^{\mathrm{*}}$ tends to the bare single-particle energy in the weak-coupling limit. Thus our theory may be viewed as an extension of the KMK theory to the strong-coupling region. The present general theory is applied to the electron gas in order to investigate whether the Coulomb interaction alone can cause superconductivity. We have improved on the two-body approximation to describe the normal state of the electron gas and have determined the effective potential variationally. With the use of the effective potential thus obtained, we have evaluated all terms up to second order for both ${V}_{p}$(k,k') and \ensuremath{\varepsilon}${\ifmmode \tilde{}\else \~{}\fi{}}_{\mathrm{k}}$. The pairing potential includes the plasmon-mediated attractive and the paramagnon-mediated repulsive parts as well as many complicated vertex corrections. We have solved the gap equation numerically and obtained the result that superconductivity appears at rather low carrier densities (i.e., ${r}_{s}$>3.9).The highest transition temperature ${T}_{c}$ in the range 30--60(${m}^{\mathrm{*}}$/${\ensuremath{\epsilon}}_{0}^{2}$) K is obtained at ${r}_{s}$\ensuremath{\approxeq}7, where ${m}^{\mathrm{*}}$ is the band mass in units of the free-electron mass and ${\ensuremath{\epsilon}}_{0}$ is the dielectric constant. Our results might be useful to help explain the mechanism of superconductivity in the high-${T}_{c}$ oxide superconductors.