Numerical infinities applied for studying Riemann series theorem and Ramanujan summation

Abstract

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid's Common Notion no. 5 'The whole is greater than the part' and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor's and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann's result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used.

Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization

“…Proof The first two relations follow immediately from (27), Table 3 and recalling that C k is nonsingular. Moreover, since β k = Ap k 2 / r k 2 , note that in (28) we have…”

“…(Notice that the noncontradictoriness of the ①-based computational methodology has been studied in depth in [15][16][17].) From the practical point of view, this methodology has given rise both to a new supercomputer patented in several countries (see [18]) and called Infinity Computer and to a variety of applications starting from optimization (see [12,[19][20][21][22][23][24]) and going through infinite series (see [13,[25][26][27][28]), fractals and cellular automata (see [25,[29][30][31][32]), hyperbolic geometry and percolation (see [33,34]), the first Hilbert problem and Turing machines (see [13,35,36]), infinite decision making processes and probability (see [13,[37][38][39]), numerical differentiation and ordinary differential equations (see [40][41][42][43]), etc.…”