From circle to square, a study on the convergence of a curve sequence

Anderson Luiz Pedrosa Porto, Douglas Frederico Guimarães Santiago, Leonardo Gomes, Márcio Henrique Marques Macedo

Abstract


In this paper we present convergence concepts applications of numerical and functions sequences to a geometric problem which raised from an analysis using GeoGebra. This question involves the traces’s family of plane curves behavior, whose initial trace of the curve is given by a circle of radius k and the whose others, intuitively, approach to a square with side measuring 2k. A proof of this convergence is done, as well a demonstration that the areas bounded by the traces of the curves and their lengths converge, respectively, to the area and length of the boundary square. We also present a brief historical setting of convergence concepts and the ideas of infinity behind these concepts. In order to be developed to graduation level, this paper goes some way meet what historically was done to determine properties of curved figures through approximations by rectilinear figures.


Keywords


Curves; Pontual convergence; Uniform convergence; Sequences; Trace

References


Ávila, G. S. S. (2011). Várias faces da matemática, 2ª Ed. Edgar Blucher, São Paulo.

Boyer, C. B. (1996). História da Matemática, 2ª Ed. Edgar Blucher, São Paulo.

Lima, E. L. (2000). Curso de Análise, vol 2, 6ª Ed. IMPA, Rio de Janeiro.

Lima, E. L. (2004). Análise real, vol 2, 1ª Ed. IMPA, Rio de Janeiro.

Lima, E. L. (2010). Análise real, vol 1, 10ª Ed. IMPA, Rio de Janeiro.




DOI: http://dx.doi.org/10.5902/2179460X32231

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