The automated home referred to as Smart Home is expected to offer fully customized services to its residents, reducing the amount of home labor, thus improving human beings' welfare. Service robots and Internet of Things (IoT) play the key roles in the development of Smart Home. The service provision with these two main components in a Smart Home environment requires: 1) learning and reasoning algorithms and 2) the integration of robot and IoT systems. Conventional computational intelligence-based learning and reasoning algorithms do not successfully manage dynamic changes in the Smart Home data, and the simple integrations fail to fully draw the synergies from the collaboration of the two systems. To tackle these limitations, we propose: 1) a stabilized memory network with a feedback mechanism which can learn user behaviors in an incremental manner and 2) a robot-IoT service provision framework for a Smart Home which utilizes the proposed memory architecture as a learning and reasoning module and exploits synergies between the robot and IoT systems. We conduct a set of comprehensive experiments under various conditions to verify the performance of the proposed memory architecture and the service provision framework and analyze the experiment results.