We consider conditions for the convergence of sequences in terms of positive and alternating Perron expansions ($P$-representation and $P^-$-representation). These conditions are crucial to determine the continuity of functions that are defined using $P$-representation or $P^-$-representation of real numbers.

翻译：本文探讨了基于正负交替佩龙展开（$P$-表示与$P^-$-表示）的序列收敛条件。这些条件对于确定以实数$P$-表示或$P^-$-表示定义的函数连续性至关重要。