Triângulo harmônico e de Leibniz

Abstract

This work aims to present theoretical material on Leibniz's harmonic triangle, its relationship with the series, the harmonic series and with Pascal's triangle, as well as to gather and bring to the knowledge of the public that appreciates the standards of mathematics, whether they are students of the high school or higher education, a basis throughout the studies for the development of such knowledge. The harmonic triangle was defined by Leibniz (1646-1716) in 1673, with a definition related to the successive differences of the harmonic series, and such a definition was possible due to the fact that Leibniz had studied several different mathematical texts throughout your life. The formation of this harmonic triangle is made by the reciprocal of the elements of Pascal's triangle times their own numbers. This harmonic triangle allows you to work with series and can even be used to calculate areas. This definition was made from the study of the harmonic series, and after analysis of its properties, used to perform the finite and infinite sums of series through a procedure called, by Leibniz, “sum of all differences”.