We report a high frequency/high magnetic field electron spin resonance (HF-ESR) spectroscopy study in the sub-THz frequency domain of the two representatives of the family of magnetic topological insulators (MnBi$_{2}$Te$_{4}$)(Bi$_{2}$Te$_{3}$)$_{n}$ with $n = 0$ and 1. The HF-ESR measurements in the magnetically ordered state at a low temperature of $T = 4$ K combined with the calculations of the resonance modes showed that the spin dynamics in MnBi$_{text{4}}$Te$_{text{7}}$ is typical for an anisotropic easy-axis type ferromagnet (FM) whereas MnBi$_{text{2}}$Te$_{text{4}}$ demonstrates excitations of an anisotropic easy-axis type antiferromagnet (AFM). However, by applying the field stronger than a threshold value $sim 6$ T we observed in MnBi$_{text{2}}$Te$_{text{4}}$ a crossover from the AFM resonance modes to the FM modes which properties are very similar to the ferromagnetic response of MnBi$_{text{4}}$Te$_{text{7}}$. We attribute this remarkably unusual effect unexpected for a canonical easy-axis AFM, which, additionally, can be accurately reproduced by numerical calculations of the excitation modes, to the closeness of the strength of the AFM exchange and magnetic anisotropy energies which appears to be a very specific feature of this compound. Our experimental data evidences that the spin dynamics of the magnetic building blocks of these compounds, the Mn-based septuple layers (SLs), is inherently ferromagnetic featuring persisting short-range FM correlations far above the magnetic ordering temperature as soon as the SLs get decoupled either by introducing a nonmagnetic quintuple interlayer, as in MnBi$_{text{4}}$Te$_{text{7}}$, or by applying a moderate magnetic field, as in MnBi$_{text{2}}$Te$_{text{4}}$, which may have an effect on the surface topological band structure of these compounds.
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