This paper investigates the frequency response curve (FRC) of the nonlinear oscillator using the harmonic balance method (HBM), focusing on the accuracy of solutions in the superharmonic resonance (SHR) region. The solutions obtained with different orders of harmonic function series are verified by the numerical continuation method. It is found that for every two-order increasing in HBM, an additional peak appears in the SHR region of FRC for the system with exclusively odd-order nonlinearities, enhancing the accuracy of results. However, higher-order HBM may not always yield higher solution accuracy in some local intervals near superharmonic resonance. The advantages and disadvantages of HBM with different harmonic function series are discussed and illustrated with two examples.