A Regularity in Ramanujan Summation Function Results for the Triangular Number Series

Abstract

The Triangular Number Series, defined as 1 + 2 + 3 + 4 + ..., when subjected to Ramanujan Summation, gives the known and somewhat controversial result of -1/12. The Ramanujan Summation Function is defined in such a way as to accept any such series and produce corresponding values for them, thus allowing for the aforementioned result to be obtained as well. In this paper, the Triangular Number Group Series Function is defined as a function that generates members of a sum, whereas the values of those members depend on a parameter g, representing the number of elements from the Triangular Number Series in each member of the new sum. This paper shows that the results of the Ramanujan Summation Function upon any such Triangular Number Group Series follow a regularity that is also dependent on the parameter g. Once such a regularity is obtained, the scope of g is extended to real and complex numbers as well.