Euler's Series Methods and Figurate Numbers
of the series $1-2+3-4+\ldots$ The results obtained following this method can then be combined together to obtain new and more complex ones. As an example, I want to report a result with relative method extracted from Hardy's book. It consists in summing the series $s = 1-2+3-4+\ldots$ For the first time this series was summed by Euler, who managed to assign the finite value $-\frac{1}{4}$ and can be useful to see how where one method fails, another can succeed. First of all we notice that the series proceeds with a succession of alternating signs, that is, it alternates a negative number with a positive number with behavior similar to Grandi's series. This can give us a starting point to involve precisely Grandi's series that we have treated previously: $\frac{1}{2} = 1-1+1-1+\ldots$ Subtracting Grandi's series from the original series $s$, we obtain: $$s - \frac{1}{2} = (1-2+3-4+\ldots) - (1-1+1-1+\ldots)$$ which in another way can be written as: $$s - \frac{1}{2} = 0-1+2-3+4-\ldots$$ Also in this case, the same result can be obtained by multiplying the series by $-1$ and therefore: $$-s = 1+2-3+4-\ldots$$ from which we obtain: $$s - \frac{1}{2} = 0-1+2-3+4-\ldots = -s$$ and therefore the result found for the first time by Euler: $$1-2+3-4+\ldots = -\frac{1}{4}$$ Comforted by these results, we might be interested in finding the value of a particularly interesting series involving all Natural Numbers, i.e.: $$1+2+3+4+5+\ldots \quad (27)$$ but as the reader will notice with a little application, the methods suggested so far are not sufficient to treat such a series for which we will have to consider the symmetries of Riemann's zeta function. Figurate Numbers and Generating Functions (Tabit) Figurate Numbers are a class of numbers that directly refers to well-defined regular geometric figures. The reason for this association is given by a symmetric characteristic inherent in the number itself. Triangular numbers, for example, are all those numbers that can be geometrically arranged in the shape of a regular triangle. Similarly, square numbers are all those numbers that can be arranged so as to geometrically represent a square. In general terms, all those numbers whose internal arrangement refers to a regular polygon are called polygonal numbers. In these cases the internal arrangement of the number follows the lines of a regular figure and this is therefore considered important for the understanding of the number and associated with its maximum and perfect expression. A triangular number, for example, will inevitably refer to the concepts expressed by the geometric figure of the triangle, a pentagonal number to that of the pentagon. In the case where a number possesses both these symmetries, both these symbolic values must be taken into consideration as significant. The habit of classifying numbers in relation to these geometric configurations with high symmetry is very ancient and considered of extreme importance. Evagrius Ponticus, in the introduction to his book On Prayer, refers to the symbolic meaning of some classes of polygonal numbers, thus testifying to an advanced use of numerical symbolism among the early Fathers of the Christian Church. He provides in his introduction a detailed set of explanations and symbolic correspondences associated with figurate numbers that we report: • Triangular numbers: the Science of the Holy Trinity, to the triangular figure correspond then Faith, Hope and Charity like gold, silver and precious stones. • Square numbers: the Quaternary of Virtues, to this correspond then incense, cassia, onyx and myrrh. • Hexagonal numbers: the line that denotes the current ordering of the world. • Spherical numbers: associated with the cyclical course of time. In this context the number 25 is indicated as a symbol of true knowledge of this century. Figure 1: Example of figurate numbers with their respective geometric representation Spherical Numbers, also called circular numbers or automorphs, are those numbers that once squared present themselves in the final part of the number. The succession of spherical numbers is $1, 5, 6, 25, 76, 376, 625, \ldots$ Figure 2: Succession of tetrahedral numbers. The general formula to derive the nth tetrahedral number is: $$T_4(n) = \frac{1}{6}n(n+1)(n+2)$$ Alongside two-dimensional figurate numbers there are also other forms of figurate numbers inspired by geometric objects with three or more dimensions. An example are Pyramidal Numbers as well as Platonic Numbers which enjoy a certain importance. • Tetrahedral numbers