**Abstract** : A systematic study is presented for centrality, transverse momentum ($p_T$) and pseudorapidity ($\eta$) dependence of the inclusive charged hadron elliptic flow ($v_2$) at midrapidity ($|\eta| < 1.0$) in Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 19.6, 27 and 39 GeV. The results obtained with different methods, including correlations with the event plane reconstructed in a region separated by a large pseudorapidity gap, and 4-particle cumulants ($v_2\{4\}$), are presented in order to investigate non-flow correlations and $v_2$ fluctuations. We observe that the difference between $v_2\{2\}$ and $v_2\{4\}$ is smaller at the lower collision energies. Values of $v_2$, scaled by the initial coordinate space eccentricity, $v_{2}/\varepsilon$, as a function of $p_T$ are larger in more central collisions, suggesting stronger collective flow develops in more central collisions, similar to the results at higher collision energies. These results are compared to measurements at higher energies at RHIC ($\sqrt{s_{NN}}$ = 62.4 and 200 GeV) and at LHC (Pb + Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV). The $v_2(p_T)$ values for fixed $p_T$ rise with increasing collision energy within the $p_T$ range studied ($< 2 {\rm GeV}/c$). We compare the $v_2$ results to UrQMD and AMPT transport model calculations, and physics implications on the dominance of partonic versus hadronic phases in the system created at BES energies are discussed.