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The Rayleigh – Schrödinger perturbation series possesses the property that can be termed multiple convergence. This property is that using appropriate multivalued approximants it enables the energy levels of several states to be calculated from the coefficient of a series constructed for one state. This multiple convergence property has been previously demonstrated in a number of papers: for linear anharmonic oscillators (Sergeev and Goodson J. Phys. A: Math. Gen. 31 4301 (1998)), for one-dimensional periodic problem (Fernandez and Diaz 2001, Eur. Phys. J. D15 41), for electronic sates of diatomic molecules (Jordan Int. J. Quant. Chem. Symp. 9, 325 (1975); Goodson Mol. Phys. 110, 1681 (2012)) and for vibrational states of formaldehyde (Duchko and Bykov Phys. Scr. 94, 105403, (2019)). In this paper, we examine multiple convergence as applied to the vibrational states of the hydrogen sulphide which exhibits the strong local mode behaviour. We show that property of multiple convergence is connected with the polyad structure of the vibrational energy spectrum and the strength of anharmonic resonance coupling between the states. [ABSTRACT FROM AUTHOR]