Recent advances in Large Language Models (LLMs) have been changing the paradigm of Recommender Systems (RS). However, when items in the recommendation scenarios contain rich textual information, such as product descriptions in online shopping or news headlines on social media, LLMs require longer texts to comprehensively depict the historical user behavior sequence. This poses significant challenges to LLM-based recommenders, such as over-length limitations, extensive time and space overheads, and suboptimal model performance. To this end, in this paper, we design a novel framework for harnessing Large Language Models for Text-Rich Sequential Recommendation (LLM-TRSR). Specifically, we first propose to segment the user historical behaviors and subsequently employ an LLM-based summarizer for summarizing these user behavior blocks. Particularly, drawing inspiration from the successful application of Convolutional Neural Networks (CNN) and Recurrent Neural Networks (RNN) models in user modeling, we introduce two unique summarization techniques in this paper, respectively hierarchical summarization and recurrent summarization. Then, we construct a prompt text encompassing the user preference summary, recent user interactions, and candidate item information into an LLM-based recommender, which is subsequently fine-tuned using Supervised Fine-Tuning (SFT) techniques to yield our final recommendation model. We also use Low-Rank Adaptation (LoRA) for Parameter-Efficient Fine-Tuning (PEFT). We conduct experiments on two public datasets, and the results clearly demonstrate the effectiveness of our approach.