The symbol \(\mathcal{S}(X)\) denotes the hyperspace of finite unions of convergent sequences in a Hausdorff space \(X\). This hyperspace is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in \(\mathcal{S}(X)\). Then we consider some cardinal invariants on \(\mathcal{S}(X)\), and compare the character, the pseudocharacter, the \(sn\)-character, the \(so\)-character, the network weight and \(cs\)-network weight of \(\mathcal{S}(X)\) with the corresponding cardinal function of \(X\). Moreover, we consider rank \(k\)-diagonal on \(\mathcal{S}(X)\), and give a space \(X\) with a rank 2-diagonal such that \(S(X)\) does not have any \(G_{\delta}\)-diagonal. Further, we study the relations of some generalized metric properties of \(X\) and its hyperspace \(\mathcal{S}(X)\). Finally, we pose some questions about the hyperspace \(\mathcal{S}(X)\).