Transposición Didáctica de las series numéricas infinitas. Una caracterización del Discurso Escolar actual en el nivel superior.

In this thesis one has looked for to find reasons on why the students face so many problems just when they initiate the study of sequences and infinite series, both numerical and of functions. The work contemplates an historical study from two thousand years before Christ to the present time. Of this form we found that for the last four thousand years mathematicians –and non mathematicians- had been generating, modifying and selecting concepts such as sequence, series, convergence, at the same time that arose and disappeared methods that allow generating examples of those concepts. We found that, from our point of view, the history and development of the infinite series can be divided in five different periods, but that in many occasions they are overlapped making difficult to find where finishes one and the following one begins. Parallel to the historical part we have made an analysis of the form in which these concepts were boarded in the classrooms, or its equivalent one, to be able to understand how we see the infinite series and how they were seen. This analysis took us to observe diverse effects of the called process Didactic Transposition and how it has been modifying our text books. Finally we made a characterization of the School Discourse that we found in our classrooms at the moment in which this work was written. Using all these results we have analyzed the form in which the students with the infinite series answer and work, at the same time that we could observe why in our opinion some investigations have not obtained the expected results for by their authors