In the present paper, we introduce and study the convergence of a sequence of closable linear operators in a Banach space. Moreover, we prove that if Tn converges in the generalized sense to T, where T and (Tn)n∈N are closed linear operators, then there exists a non negative integer element n0 such that, for all n≥n0, we have the Weyl essential spectrum of Tn included in the Weyl essential spectrum of T (see Theorem 3.1). The same study is made for the case of convergence to zero compactly under which weaker results are established (Theorem 3.3 and Corollary 3.1).