The rate of convergence for an almost certainly convergent seriesindependent random elements in a real separable Banach space is studied in this paper. More specifically, whenSnconverges almost certainly to a random elementS, the tail seriesis a well defined sequence of random elements withalmost certainly. The main result establishes for a sequence of positive constantswithbj⩽ Const.bnwheneverthe equivalence between the tail series weak law of large numbersand the limit lawthereby extending a result of Nam and Rosalsky [20] to a Banach space setting while also simplifying the argument used in the earlier result. The quasimonotonicity proviso oncannot be dispensed with