Autonomous robots operating in open environments need the ability to continuously handle tasks that are not covered by predefined local methods. However, existing approaches often rely on repeated large-language-model (LLM) interaction for uncovered tasks, and even successful executions or observed successful external behaviors are not always autonomously transformed into reusable local knowledge. In this paper, we propose an LLM-driven closed-loop autonomous learning framework for robots facing uncovered tasks in open environments. The proposed framework first retrieves the local method library to determine whether a reusable solution already exists for the current task or observed event. If no suitable method is found, it triggers an autonomous learning process in which the LLM serves as a high-level reasoning component for task analysis, candidate model selection, data collection planning, and execution or observation strategy organization. The robot then learns from both self-execution and active observation, performs quasi-real-time training and adjustment, and consolidates the validated result into the local method library for future reuse. Through this recurring closed-loop process, the robot gradually converts both execution-derived and observation-derived experience into reusable local capability while reducing future dependence on repeated external LLM interaction. Results show that the proposed framework reduces execution time and LLM dependence in both repeated-task self-execution and observation-driven settings, for example reducing the average total execution time from 7.7772s to 6.7779s and the average number of LLM calls per task from 1.0 to 0.2 in the repeated-task self-execution experiments.