Acelerando a convergência da série de Taylor de funções elementares : um método baseado em frações contínuas

Abstract

At Taylor series elementary functions are applied two algebric methods which converts them into continued fractions. Euler's method causes the convergents of this continued fraction to be exactly equal to partial sums of the series that originated it. Already the convergents of the continued fraction generated by the sucessive substitution method are a rational approximate for the referred function. A contraction process is applied to the continued fractions originated by these methods, which results in new continued fractions, characterized by converges more quickly to the value of the function than the own series. Graphic and numeric comparisons between Taylor series of the function, the continued fractions generated by the methods and its contractions are performed. It is observed that the convergents of the even contraction order 5 of the continued fractions obtained by the sucessive substitution method results, in average, approximately with error in the order of 10-8 of the real value of the analized functions, rate that can be considered very good when it is compared with Taylor's polinomials value of the same order. The described methods have complementary characteristics, which assign to the contraction of its continued fractions a possible and eficient algorithm implementation.