A tuned mass damper (TMD) system generates structural control forces through large motions of mass units. Therefore, it may not be functional if the stroke capacity of its spring and damper components are insufficient. This paper focuses on a novel mass-damper system, the series multiple tuned mass damper (SMTMD) system, which consists of multiple interconnected TMDs, of which only the first is connected to the primary structure. The main purpose of this paper is to compare the control effectiveness and TMD stroke of SMTMDs with those of a conventional TMD device. In addition, the ability of the studied SMTMD to suppress multiple structural modes is also investigated. First, the optimal design theory for an SMTMD installed on an arbitrary floor of a multi-storey building is developed. To optimize the SMTMD parameters, two performance indices are established by combining multiple modal responses of the primary structure. The developed theory is demonstrated analytically by using a three-story building. The results show that the SMTMD with a higher number of TMD units places lower demands on the TMD stroke and is more adaptive in controlling multiple structural modes of the primary structure.