\textit{E. T. Copson} [Proc. Edinb. Math. Soc., II. Ser. 17, 159-164 (1970; Zbl 0223.40001)], showed that a bounded positive sequence \(\{a_ n\}\) satisfying \(a_{n+r}\leq \sum_{s=1}^ r k_ s a_{n+r-s}\), \(k_ s>0\), \(k_ 1+ \cdots+ k_ r =1\) \(\forall n\) is necessarily convergent. \textit{C. Rossi} [Monotonia alle Copson e sue generalizzazioni. Tesi di Laurea, Università di Modena (1990)] generalized Copson's theorem for the case \(r=2\), replacing the above condition by \(a_{n+2}\leq (1- \lambda_{n+2}) a_ n+ \lambda_{n+2} a_{n-1}\), \(0<\Lambda_ 1\leq \lambda_ n\leq \Lambda_ 2< 1\) \(\forall n\). The present authors define a sequence \(\{a_ n\}\) to be \((\alpha_ n, \lambda_ n)\)-recurrent if \(0< a_ n\leq M\) \(\forall n\) and \(\exists \{\lambda_ n\}\), \(\{\alpha_ n\}\) with \(0\leq \lambda_ n <1\) and \(0< \alpha_ n\leq 1\) such that \(a_ n= \alpha_ n [(1- \lambda_ n) a_{n-2}+ \lambda_ n a_{n-1}]\) \(\forall n\geq 3\) and \(a_ 1\neq a_ 2\), and \(\lambda_ n\)- recurrent if all \(\alpha_ n =1\). In the first section it is shown that a \(\lambda_ n\)-recurrent sequence is convergent if and only if \(\sum \lambda_ n\) is divergent. Various conditions on \((\alpha_ n, \lambda_ n)\)-recurrent sequences \(\{a_ n\}\), too complicated to summarize nicely here, are shown in section 2 to imply that \(\{a_ n\}\) is convergent, is a null sequence, has at most two cluster points, or may have a preassigned countable set of cluster points, and some illustrative examples are given. In the final two sections some classical results on series, based on the convergence properties of monotone sequences, are shown to continue to hold for \((\alpha_ n, \lambda_ n)\)-recurrent or \(\lambda_ n\)-recurrent sequences. Included here is an extension of Leibniz's theorem on alternating series.