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Três problemas sobre série harmônica na Olimpíada Internacional de Matemática

2020; Lázaro, C. and Rodrigues, T.; Volume: 17; Linguagem: Inglês

10.21167/cqdvol17ermac202023169664jllabagfb127138

2316-9664

Juan López Linares, Alexys Bruno‐Alfonso, Grazielle Feliciani Barbosa,

Chemistry Education and Research

The International Mathematical Olympiad (IMO) is the most prestigious pre-university math competition.In this article we study in detail three problems proposed in different years and that somehow use the harmonic series.They can be incorporated into the training of high school students for this type of contention or in classes at college level.The first one was proposed for the 2001 IMO and deals with an inequality, that must be shown to be valid for an infinite number of positive integers.The solution considers the fact that the harmonic series grow unlimited.The second one was proposed for the IMO of 1979 and chosen as the first tournament question.In this case the focus is on a partial sum of the alternating harmonic series.A generalization of this problem is found.The third one was proposed for the IMO of 1975 and deepens the understanding of the harmonic series.Taking away an infinite subsequence does the harmonic series remains divergent?