Divergent Series Manipulation and the Functional Method
The manipulation of these series is indeed very delicate and can lead to algebraic absurdities if one does not proceed with due caution and using only the operations that are permitted. In these series, for example, it is not possible to reorganize the terms of the series by summing some first and then others. Specifically, the direct algebraic manipulation of divergent series allows with certainty only three algebraic manipulations, which represent the minimum conditions that every summation method must satisfy4 to be coherent:
1. Multiplying all terms of the series $\{a_n\}$ by the same coefficient $k$ is equivalent to multiplying the sum of the series by the same coefficient, i.e.,
$$\sum_{n=0}^{\infty} a_n = s \Rightarrow \sum_{n=0}^{\infty} ka_n = ks \quad (18)$$2. The first term of the series can be subtracted from both sides of the equivalence, i.e.,
$$a_0 + \sum_{n=1}^{\infty} a_n = s \Rightarrow \sum_{n=1}^{\infty} a_n = s - a_0 \quad (19)$$3. Summing term by term two distinct series $\{a_n\}$ and $\{b_n\}$ produces the sum of the two series, i.e.,
$$\left\{\begin{array}{l} \sum_{n=0}^{\infty} a_n = s \\ \sum_{n=0}^{\infty} b_n = t \end{array}\right\} \Rightarrow \sum_{n=0}^{\infty} (a_n + b_n) = s + t \quad (20)$$The Functional Method
The preceding algebraic manipulations allow us to introduce an interesting and elegant method which is the direct summation method, also called functional. Suppose we want to find the value $s$ of the following divergent series:
$$2 + 4 + 8 + 16 + 32 + \cdots = s \quad (21)$$We can note that, factoring out a factor 2 from each term of the series, we have:
$$2 + 4 + 8 + 16 + \cdots = 2(1 + 2 + 4 + 8 + \cdots) \quad (22)$$that is, we obtain the same series multiplied by a factor 2 and to which the term 1 has been added, i.e.,
$$s = 2(1 + s) \quad (23)$$But the preceding equation is a simple first-degree equation which, once solved, leads to determining
$$s = -2 \quad (24)$$and, consequently, brings us to the result
$$2 + 4 + 8 + 16 + 32 + \cdots = -2 \quad (25)$$We notice the symbolic perfection of the series we have found: the sum of the powers of 2, while developing phenomenally in its powers, adds nothing qualitatively to the Number 2. The result of the summation always remains the same progenitor 2, with the minus sign which, as in the preceding cases, indicates the hidden or unmanifest action of such Number.
At this point we can apply ourselves to the study of another notable series: Grandi's series.
$$1 - 1 + 1 - 1 + \cdots$$To apply the functional method to this series we must assign it a value, which we will identify with the letter $s$, i.e.,
$$s = 1 - 1 + 1 - 1 + \cdots$$and seek a symmetry of the preceding expression. We note that if we add the number -1 to both members of the series, we obtain the following relation:
$$s - 1 = -1 + 1 - 1 + \cdots$$whose series expressed in the right side of the expression is identical to what would be obtained by multiplying the original series by $-1$, i.e.,
$$-s = -1 + 1 - 1 + \cdots$$Here, then, is the symmetry we were looking for: Grandi's series provides the same result under multiplication or addition of -1, i.e.,
$$s - 1 = -1 + 1 - 1 + \cdots = -s$$which in other terms means
$$s - 1 = -s$$and consequently
$$s = \frac{1}{2}$$We can therefore write that
$$1 - 1 + 1 - 1 + \cdots = \frac{1}{2}$$Also in this case the symbolism of the series is evident. The indefinite polarization of the positive and negative, in their phenomenal alternation, is summarized by the dualism expressed by $1/2$.
Summation of Geometric Series
With the functional method one can easily find the sum of any geometric series, i.e., reduce to the following formula:
$$1 + n + n^2 + n^3 + \cdots = \frac{1}{1-n}$$The demonstration is similar to the preceding and is based on a fundamental symmetry of the geometric series which is invariant under scale transformation. Indeed, let us call the sum of the series $s$, i.e.,
$$s = 1 + n + n^2 + n^3 + \cdots$$and multiply both sides of the expression by the factor $n$, we obtain
$$ns = n \cdot (1 + n + n^2 + n^3 + \cdots)$$i.e.,
$$ns = n + n^2 + n^3 + n^4 + \cdots$$Similarly, if we subtract 1 from the starting series, we obtain
$$s - 1 = n + n^2 + n^3 + n^4 + \cdots$$and, therefore,
$$ns = n + n^2 + n^3 + n^4 + \cdots = s - 1$$which means
$$ns = s - 1$$and consequently
$$s = \frac{1}{1-n}$$With this notable result we can obtain numerous interesting formulas from a symbolic point of view. We want to report one that is in some way connected with the article we presented in this same journal and dedicated to 'Unitotality and the birth of Numbers'. Indeed, if we consider $n = 10$ and write explicitly the divergent geometric series associated with it, we obtain:
$$1 + 10 + 100 + 1000 + \cdots$$With the preceding result we can think of reducing this series to a finite qualitative value and here it is:
$$1 + 10 + 100 + 1000 + \cdots = -\frac{1}{9}$$i.e.,
$$1 + 10 + 100 + 1000 + \cdots = -0.111111\ldots$$If we now want to consider its square we obtain an expression similar to what we made explicit in the first article, i.e.,
$$(1 + 10 + 100 + 1000 + \cdots)^2 = 0.0123456789\ 0123456789\ldots \quad (26)$$