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There are typically several <i>perturbation methods</i> for approaching the solution of <i>weakly nonlinear vibrations</i> (where the nonlinear terms are “small” compared to the linear ones): the <i>Method of Strained Parameters</i>, the <i>Naive Singular Perturbation Method</i>, the <i>Method of Multiple Scales</i>, the <i>Method of Harmonic Balance</i> and the <i>Method of Averaging</i>. The <i>Straightforward Expansion Perturbation Method</i> (SEPM) applied to weakly nonlinear vibrations does not usually yield to correct solutions. In this manuscript, we provide mathematical proof of the inaccuracy of the SEPM in general cases. Nevertheless, we also provide a sufficient condition for the SEPM to be successfully applied to weakly nonlinear vibrations. This mathematical formalism is written in the syntax of the first-order formal language of Set Theory under the methodology framework provided by the Category Theory.