The convergence of sequences in terms of positive and alternating Perron expansions

In this paper we study conditions for the convergence of sequences in terms of positive and alternating Perron expansions of real numbers (P-representation and P−-representation). These conditions make it possible to verify the continuity of functions defined via the P-representation or P−-representation. The main results of this work are as follows. First, we systematically consider all possible cases and, by taking into account the geometric structure of Perron expansions, significantly reduce the number of distinct cases that require separate analysis. Second, for each case, we formulate and rigorously prove a convergence condition that is both necessary and sufficient. Third, the results are obtained not for specific expansions (such as the Engel or Pierce expansions), but for the Perron expansions, which include many classical representations of real numbers as special cases. These results provide a foundation for further analysis of fractal functions defined through Perron expansions or their special cases (the Lüroth, Engel, Pierce, and Sylvester expansions).