Sequências e séries de funções: aspectos teóricos e propostas didáticas ao ensino de cálculo

Abstract

The present work offers a comprehensive study of numerical sequences, numerical series, sequences of functions, and series of functions, articulating the foundations of Real Analysis with methodological proposals aimed at teaching these topics in introductory Calculus courses. Initially, formal definitions, fundamental properties, and classical criteria of convergence are discussed, with the purpose of establishing a solid and coherent conceptual basis. From this, in order to mitigate or eliminate possible difficulties encountered by students in understanding these subjects, the work proposes didactic approaches that make use of manipulable materials, visual representations, and technological resources, seeking to promote meaningful learning and reduce the obstacles present in traditional instruction. Such methodologies allow students to visualize patterns, establish conceptual relationships, and move more confidently between concrete and formal representations, contributing to a deeper understanding of the mathematical phenomena involved. The analysis shows that the integration between theoretical rigor and innovative teaching strategies constitutes an effective path to overcoming recurring difficulties and fostering student engagement. In conclusion, it is stated that the use of concrete materials, explanatory figures, and support software can enhance the teaching of sequences and series, making it more accessible, intuitive, and coherent with the needs of contemporary education. The study also points to future research perspectives involving the practical application of the proposed approaches and their evaluation in real classroom contexts, contributing to the continuous improvement of pedagogical practices in Mathematics education.