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During radiation-dominated X-ray transients in AGNs, energy is rapidly transferred between the photons and the electrons until the electron temperature ($T_e$) has equilibrated to the inverse-Compton temperature of the radiation ($T_{\rm IC}$). This occurs on a much shorter timescale than the equilibration of the radiation into a Wien distribution. Once the temperatures have equilibrated, the condition $T_e=T_{\rm IC}$ continues to be enforced by the Compton exchange of energy between the photons and the electrons. The radiative transfer problem then becomes highly nonlinear, since the variation of the electron temperature determines the spectrum (through the Kompaneets equation), while the spectrum determines the electron temperature (through the condition $T_e=T_{\rm IC}$). It is therefore extremely useful to develop techniques for independently calculating the electron temperature as a function of time for an arbitrary initial radiation spectrum. In this paper we develop a general technique for determining the electron temperature by using the frequency moments of the initial spectrum to construct a continued fraction with dramatic convergence properties. Resulting expressions for $T_e$ are presented for a variety of initial spectra.
