We study the edge modes of a finite LC-resonator circuit that consists of alternatingly arranged inductors, separated by identical capacitors, each of which is grounded. The circuit has the configuration of the Su-Schrieffer-Heeger (SSH) model, and one-to-one correspondence between these two models can be established. Interestingly, when the corresponding SSH model is in the topological nontrivial regime, only one topological edge mode appears in the finite LC-resonator circuit, and the other is absent. Through the analysis of the circuit Hamiltonian, we find that only by reducing the boundary capacitance can the missing topological edge mode occur, but the reappeared topological mode is different from the standard one. When the boundary capacitance is lower than a critical value, a Tamm mode related to the local resonance also appears. When the finite circuit system is in the trivial regime, the replacement of the boundary capacitor could result in only the Tamm mode due to the restriction of the bulk-edge correspondence. The present study of the influence of the boundary configuration on the topological edge modes will deepen our understanding of topolectrical circuits.