Infinite Limit of Sequences and Its Phenomenology

Abstract

In this document, we search for and define an infinite limit of sequences that is correct and accepted by the mathematical experts, the final purpose of which is to analyze its phenomenology, in Freudenthal's sense. To make the choice, experts were consulted on two issues. The first one was not decisive because of the effect that the divergence term causes, and for this reason, we did a second expert consultation where this term was removed and we selected the definition we have analyzed in this document. Once the definition was chosen, two approaches were considered for analysis: the intuitive approach and the formal approach. Based on these two approaches, we specify certain phenomena organized by the definition: unlimited intuitive growth and unlimited intuitive decrease (intuitive approach) and one way and return infinite limit of sequences (formal approach), and show examples of such phenomena by graphical, verbal and tabular representation systems. All this aim to be a help to overcome the difficulties that pre-university students have with the concept of limit.

Infinite limit of a function at infinity and its phenomenology

“…The sequence is not subject to an upper limit and seems to grow in an unlimited way. This allows us to deduce that the sequence has a limit +∞ (Arnal-Palacián et al, 2020).…”

“…In this study we took the one used by Linés (1983) in the manual Principles of Mathematical Analysis. This manual was referenced in the research conducted by Arnal-Palacián (2019) and Arnal-Palacián et al (2020). Since the present research is based on the previous study, it is precisely the definition given in this manual that is considered for the present study.…”

“…We start at a real number, for example H = 49 (Figure 3), situated at the Axis Y and go "one way" to a natural number, in our example v = 7, situated at the Axis X, and "return" from n = 8(n ≥ v) to a real number of the sequence, a 8 = 64(a n > H) (Arnal-Palacián et al, 2020).…”