A model for La₁₋ₓSrₓMnO₃ incorporating dynamic Jahn-Teller and double exchange effects is presented and solved using a dynamical mean field approximation. The model explains the resistivity and magnetic transition temperature behavior of La₁₋ₓSrₓMnO₃ in the doping range 0.2 ≤ x ≤ 0.4. The material is a ferromagnetic metal at low temperatures and a paramagnet at high temperatures, with a sharp drop in resistivity near the transition temperature T_c(x). The colossal magnetoresistance observed in this material is attributed to the interplay of these two effects. The model considers the Mn d-orbitals as the electronically active orbitals, with a mean of 4-x d-electrons per Mn. The cubic anisotropy and Hund's rule coupling lead to a core spin S_c of magnitude 3/2, while the remaining electrons form a band of width ~2.5 eV. The large Hund's rule coupling J_H affects the hopping of outer shell electrons between Mn sites, being maximal when the core spins are parallel and minimal when antiparallel. This is known as double exchange, which is considered the main physics in the doping range 0.2 ≤ x ≤ 0.5. However, double exchange alone cannot explain the large resistivity of the T > T_c phase or the sharp drop in resistivity just below T_c. The necessary extra physics is a strong electron-phonon coupling due to a Jahn-Teller splitting of the Mn d⁴ state. The competition between electron itineracy and self-trapping is controlled by the dimensionless ratio of the Jahn-Teller self-trapping energy E_J-T and an effective hopping matrix element t_eff. When E_J-T/t_eff exceeds a critical value, a crossover from a Fermi liquid to a polaron regime occurs. The model Hamiltonian includes terms for electronic and Jahn-Teller effects, with the electronic part neglecting on-site Coulomb interactions. The solution involves a dynamical mean field approximation, leading to a self-consistent equation for the effective field G_eff(ω). The results show a second order phase transition at T_c = t/12λ², separating two insulating phases with slightly different gaps. The resistivity calculations show a sharp drop below T_c and a dependence on magnetic field, consistent with the observed colossal magnetoresistance. The model is consistent with experimental observations, including the shift in T_c and resistivity anomaly with doping. The results also suggest that the interplay of polaron and double exchange physics is essential for the observed behavior. The model is extended to study structural transitions and charge ordered phases at low x and x ≈ 0.5, requiring further exploration of the mean field theory.A model for La₁₋ₓSrₓMnO₃ incorporating dynamic Jahn-Teller and double exchange effects is presented and solved using a dynamical mean field approximation. The model explains the resistivity and magnetic transition temperature behavior of La₁₋ₓSrₓMnO₃ in the doping range 0.2 ≤ x ≤ 0.4. The material is a ferromagnetic metal at low temperatures and a paramagnet at high temperatures, with a sharp drop in resistivity near the transition temperature T_c(x). The colossal magnetoresistance observed in this material is attributed to the interplay of these two effects. The model considers the Mn d-orbitals as the electronically active orbitals, with a mean of 4-x d-electrons per Mn. The cubic anisotropy and Hund's rule coupling lead to a core spin S_c of magnitude 3/2, while the remaining electrons form a band of width ~2.5 eV. The large Hund's rule coupling J_H affects the hopping of outer shell electrons between Mn sites, being maximal when the core spins are parallel and minimal when antiparallel. This is known as double exchange, which is considered the main physics in the doping range 0.2 ≤ x ≤ 0.5. However, double exchange alone cannot explain the large resistivity of the T > T_c phase or the sharp drop in resistivity just below T_c. The necessary extra physics is a strong electron-phonon coupling due to a Jahn-Teller splitting of the Mn d⁴ state. The competition between electron itineracy and self-trapping is controlled by the dimensionless ratio of the Jahn-Teller self-trapping energy E_J-T and an effective hopping matrix element t_eff. When E_J-T/t_eff exceeds a critical value, a crossover from a Fermi liquid to a polaron regime occurs. The model Hamiltonian includes terms for electronic and Jahn-Teller effects, with the electronic part neglecting on-site Coulomb interactions. The solution involves a dynamical mean field approximation, leading to a self-consistent equation for the effective field G_eff(ω). The results show a second order phase transition at T_c = t/12λ², separating two insulating phases with slightly different gaps. The resistivity calculations show a sharp drop below T_c and a dependence on magnetic field, consistent with the observed colossal magnetoresistance. The model is consistent with experimental observations, including the shift in T_c and resistivity anomaly with doping. The results also suggest that the interplay of polaron and double exchange physics is essential for the observed behavior. The model is extended to study structural transitions and charge ordered phases at low x and x ≈ 0.5, requiring further exploration of the mean field theory.