Classical mathematical analysis involves an intriguing interplay between finite and infinite collections and between discrete and continuous structures. What makes the interplay intriguing is the emergence from the most elementary considerations of results that are paradoxical, or, at least, utterly counterintuitive. What, for example is the sum of $1+2+3+\text{\u2026}+n+\text{\u2026}$ ?
The sum?
Surely this sequence of numbers is simply getting bigger and bigger; and beyond this, it is not converging to anything.
Not so. The sum is –1/12.
Niels Abel called divergent series the Devil’s invention.^{1} Having made outstanding contributions to the subject as the Devil’s plaything, he knew what he was talking about.
Infinite Sequences
Mathematical analysis begins by considering infinite sequences of real numbers and by defining their limit. For the moment, I will consider the real numbers and only the real numbers. An infinite sequence is an arbitrary function whose domain coincides with the set $\mathbb{N}=\left\{0,1,2,\text{\u2026},n,\text{\u2026}\right\}$ of natural numbers, or with some of its infinite subsets, and whose range is contained in the real line $\mathbb{R}$. Thus $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ or more explicitly, if less compactly
${a}_{0},{a}_{1},{a}_{2},\text{\u2026},{a}_{n},\text{\u2026}$.
The terms ${a}_{n}$ of such sequences are often generated by a fixed rule, or by applying a specific algorithm to each $n$.
Limits are next. A real number $a$ is a limit of a sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ if, for any $\epsilon >0$, there exists a natural number ${n}_{0}={n}_{0}\left(\epsilon \right)$ such that $\left{a}_{n}a\right<\epsilon $ whenever $n>{n}_{0}$. If this is the case, the sequence converges (or tends) to $a$ as $n$ tends to infinity: $a=\mathrm{l}\mathrm{i}\mathrm{m}\left\{{a}_{n}:n\in \mathbb{N}\right\}$. It is easy to show that $a$ is unique.^{2}
The definition of a limit having been given—by AugustinLouis Cauchy, as it happens—mathematicians wished naturally to know whether and under what conditions such limits existed. The classical criterion—by Cauchy again, as it happens again—affirms that $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ is convergent if, and only if, for any $\epsilon >0$ there exists a natural number ${n}_{0}={n}_{0}\left(\epsilon \right)$ such that $\left{a}_{n}{a}_{m}\right<\epsilon $ whenever $n>{n}_{0}$ and $m>{n}_{0}$.^{3}
Among the glories of mathematical analysis, no one beyond Cauchy has ever found these ideas easy, and no one beyond Leonhard Euler has ever found them obvious.
The Cauchy criterion reflects the completeness property of the real numbers, from which necessary and sufficient condition for the existence of a limit follow almost at once. Many other conditions for convergence are more convenient in practice. If a sequence of real numbers is monotone and bounded, then it necessarily has a limit.
If two sequences $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ and $\left\{{b}_{n}:n\in \mathbb{N}\right\}$ converge to $a$ and $b$ respectively, then, for any real $r$ and $t$, the sequence $\left\{{ra}_{n}+t{b}_{n}:n\in \mathbb{N}\right\}$ converges to $ra+tb$.
Suppose that, for two sequences $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ and $\left\{{b}_{n}:n\in \mathbb{N}\right\}$, two natural numbers $m$ and $n$ can be found such that ${a}_{m+i}={b}_{n+i}$ whenever $i\in \mathbb{N}$. Then $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ converges if and only if $\left\{{b}_{n}:n\in \mathbb{N}\right\}$ converges and, in this case, so does the equality
$\mathrm{l}\mathrm{i}\mathrm{m}\left\{{a}_{n}:n\in \mathbb{N}\right\}=\mathrm{l}\mathrm{i}\mathrm{m}\left\{{b}_{n}:n\in \mathbb{N}\right\}$
The convergence of a sequence of real numbers depends only on its tail.
Example 1.
The sequence $1,{2}^{1/2},{3}^{1/3},\text{\u2026},{n}^{1/n},\text{\u2026}$ converges and its limit is equal to 1.
Example 2.
The sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$, where ${a}_{n}={\left(1+1/n\right)}^{n}$ for $n>0$, converges to Napier’s constant, e = 2.71828… .
Example 3.
The sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$, where ${a}_{n}=1+1/2+\text{\u2026}+1/n\mathrm{ln}\left(n\right)$ for $n>0$, converges to Euler’s constant, C = 0.57721… .
If there exists no limit of a sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$, then the sequence is divergent. In some sense, most real sequences are divergent.
Example 4.
Any unbounded sequence is divergent; in particular, the unbounded sequence $\left\{{b}_{n}:n\in \mathbb{N}\right\}$, where ${b}_{n}=1+1/2+\text{\u2026}+1/n$ for $n>0$.
Example 5.
The sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$, where ${a}_{n}=1$ for all even $n$, and ${a}_{n}=1$ for all odd natural $n$, is divergent.
Example 6.
The sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$, where ${a}_{n}={(n!)}^{1/n}$ for $n>0$, is unbounded, and thus divergent.
Sometimes it is convenient to treat certain divergent sequences as if they were converging to infinity. A sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ tends to +∞ if, for every $r>0$, there exists an ${n}_{0}={n}_{0}\left(r\right)$ such that ${a}_{n}>r$ whenever $n>{n}_{0}$. A sequence $\left\{{b}_{n}:n\in \mathbb{N}\right\}$ tends to –∞ if, for every $r<0$, there exists an ${n}_{0}={n}_{0}\left(r\right)$ such that ${b}_{n}<r$ whenever $n>{n}_{0}$.
Given a sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$, one can associate another sequence $\left\{{s}_{n}:n\in \mathbb{N}\right\}$ to it, where ${s}_{n}=\mathrm{\Sigma}\left\{{a}_{k}:k\le n\right\}$ denotes the sum of the first $n+1$ terms of $\left\{{a}_{n}:n\in \mathbb{N}\right\}$. Observe that the initial sequence $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ is completely determined by $\left\{{s}_{n}:n\in \mathbb{N}\right\}$, because ${a}_{n}={s}_{n}{s}_{n1}$, a point valid for all $n\in \mathbb{N}$. The pair $\left(\left\{{a}_{n}:n\in \mathbb{N}\right\},\left\{{s}_{n}:n\in \mathbb{N}\right\}\right)$ is called an infinite series, a series for short.^{4} With far less by way of notational clutter, mathematicians write
${a}_{0}+{a}_{1}+{a}_{2}+\text{\u2026}+{a}_{n}+\text{\u2026}$ .
A series ${a}_{0}+{a}_{1}+{a}_{2}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is convergent if the corresponding sequence $\left\{{s}_{n}:n\in \mathbb{N}\right\}$ has a finite limit $s$, the sum of the series; otherwise, it is divergent.
Cauchy’s convergence criterion now follows: an infinite series has a sum if and only if, for any $\epsilon >0$, there exists a ${n}_{0}={n}_{0}\left(\epsilon \right)$ such that $\left{a}_{n+1}+{a}_{n+2}+\text{\u2026}+{a}_{n+m}\right<\epsilon $ whenever $n>{n}_{0}$ and $m\in \mathbb{N}$.
The convergence of ${a}_{0}+{a}_{1}+{a}_{2}+\text{\u2026}+{a}_{n}+\text{\u2026}$ implies that
$\mathrm{lim}\left\{{a}_{n}:n\in \mathbb{N}\right\}=0$.
In the field, where time is short, simpler tests are used: D’Alembert’s test, the nth root test, Leibniz’s test, Raabe’s test, Kummer’s test, Bertrand’s test, Gauss’s test, the integral test, Abel’s test, Dirichlet’s test, Ermakov’s test.^{5}
There are sufficiently many tests to hold an endless series of undergraduate examinations.
Example 7.
The series $1/{1}^{2}+1/{2}^{2}+1/{3}^{2}+\text{\u2026}+1/{n}^{2}+\text{\u2026}$ converges, because all of its terms $1/{n}^{2}$, where $n>1$, are less than $1/\left(n\left(n1\right)\right)=1/(n1)1/n$, and the series
$1+1/2+1/6+1/12+\text{\u2026}+1/\left(n\left(n1\right)\right)+\text{\u2026}$
converges to 2. Here the socalled comparison test does its work. A more delicate calculation shows that the sum of $1/{1}^{2}+1/{2}^{2}+1/{3}^{2}+\text{\u2026}+1/{n}^{2}+\text{\u2026}$ is equal to ${\pi}^{2}/6$.^{6} This is a result well hidden from anyone’s intuitions.
As one might expect, series come in special types. There are geometric series, harmonic series, hypergeometric series, alternating series, and many more besides. In many cases, a series of the form
${a}_{0}\left(x\right)+{a}_{1}\left(x\right)+{a}_{2}\left(x\right)+\text{\u2026}+{a}_{n}\left(x\right)+\text{\u2026}$ ,
is up for grabs, where ${a}_{n}\left(x\right)\left(n\in \mathbb{N}\right)$ are realvalued functions of a real variable $x$. For each value $r$ of $x$, one has the usual series
${a}_{0}\left(r\right)+{a}_{1}\left(r\right)+{a}_{2}\left(r\right)+\text{\u2026}+{a}_{n}\left(r\right)+\text{\u2026}$ .
Did I mention power series, Taylor series, Laurent series, Dirichlet series, trigonometric series, Fourier series, and the everpopular binomial series?
Example 8.
The geometric series $1+{q}^{1}+{q}^{2}+\text{\u2026}+{q}^{n}+\text{\u2026}$ , where $\leftq\right<1$.
Example 9.
Grandi’s series: $11+11+\text{\u2026}+{\left(1\right)}^{n1}+\text{\u2026}$ and, more generally, for any strictly positive integer $k$, the series $1{k}^{1}+{k}^{2}{k}^{3}+\text{\u2026}+{\left(1\right)}^{n1}{k}^{n1}+\text{\u2026}$ .
Example 10.
The series $0!1!+2!3!+\text{\u2026}+{\left(1\right)}^{n}n!+\text{\u2026}$ .
Example 11.
Dirichlet’s series:
$f\left(x\right)={a}_{1}/{1}^{x}+{a}_{2}/{2}^{x}+{a}_{3}/{3}^{x}+\text{\u2026}+{a}_{n}/{n}^{x}+\text{\u2026}$ ,
where $x\in \mathbb{R}$, and $\left\{{a}_{n}:n\in \mathbb{N}\right\}$ is a sequence of real numbers. In particular, the generalized harmonic series, which is a part of Riemann’s zeta fuction, appears almost everywhere:
$\zeta \left(x\right)=1/{1}^{x}+1/{2}^{x}+1/{3}^{x}+\text{\u2026}+1/{n}^{x}+\text{\u2026}$ ,
and the generalized alternating harmonic series, which is a part of Dirichlet’s eta function
$\eta \left(x\right)=1/{1}^{x}1/{2}^{x}+1/{3}^{x}\text{\u2026}+{(1)}^{n1}/{n}^{x}+\text{\u2026}$ ,
is obtained in this way. The series $1/{1}^{x}+1/{2}^{x}+1/{3}^{x}+\text{\u2026}+1/{n}^{x}+\text{\u2026}$ converges for all real $x>1$, and the series $1/{1}^{x}1/{2}^{x}+1/{3}^{x}\text{\u2026}+{(1)}^{n1}/{n}^{x}+\text{\u2026}$ converges for all real $x>0$.
Example 12.
A series ${a}_{0}+{a}_{1}+{a}_{2}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is called absolutely convergent if the series $\left{a}_{0}\right+\left{a}_{1}\right+\left{a}_{2}\right+\text{\u2026}+\left{a}_{n}\right+\text{\u2026}$ converges. Every absolutely convergent series converges.
The converse is not true, alas. The alternating harmonic series
$11/2+1/31/4+\text{\u2026}+{(1)}^{n1}(1/n)+\text{\u2026}$ ,
converges and its sum is equal to $\mathrm{l}\mathrm{n}\left(2\right)$,^{7} but the same series is not absolutely convergent, as Example 4 might indicate.
A convergent series ${a}_{0}+{a}_{1}+{a}_{2}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is conditionally convergent if its associated series $\left{a}_{0}\right+\left{a}_{1}\right+\left{a}_{2}\right+\text{\u2026}+\left{a}_{n}\right+\text{\u2026}$ diverges.
If a series converges conditionally, then for any real $r$ there exists a permutation of its terms such that the new series converges to $r$. This is Riemann’s classical theorem, which admits a nontrivial generalization to a series of vectors in a finitedimensional Euclidean space.
Strangely Divergent Series
Strangely divergent series such as $1+2+3+4+\text{\u2026}$ are straightforward inasmuch as they are divergent; they are strange inasmuch as they are not. This is a claim that would seem to tiptoe to the very edge of paradox. Nonetheless, Euler dealt systematically with strangely divergent series; he was able to manipulate them successfully. After many clever manipulations, Euler thought to claim that any divergent series must have a certain sum. Euler’s position was criticized by the fastidious Jean D’Alembert, the equally fastidious Abel, and by other mathematicians, all of them indifferent to the possibility that Euler might have been right and they wrong. Only much later was some rational fragment of Euler’s idea transformed into a rich theory of the summability of divergent series.
Many divergent series; and so many sums; many sums, and so many summability methods: Abel’s summation method, Borel’s summation method, Cesaro’s summation method, Euler’s summation method, Holder’s summation method, Kummer’s summation method, Lambert’s summation method, Lindelof’s summation method, Poisson’s summation method, Voronoy’s summation method, and many others.^{8}
The simplest method of summation is named after Ernesto Cesaro. If a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is given, then it may sometimes happen—not always, not in every case—that the corresponding sequence of arithmetical means of its partial sums
${s}_{0},\left({s}_{0}+{s}_{1}\right)/2,({s}_{0}+{s}_{1}+{s}_{2})/3,\text{\u2026}$ ,
$({s}_{0}+{s}_{1}+\text{\u2026}+{s}_{n})/(n+1),\text{\u2026}$ ,
converges to a certain real number. This is its Cesaro sum.
The notion of a Cesaro sum is compatible with the usual notion of the sum of a convergent series. Indeed, according to one of Cauchy’s classical theorems, if a sequence of reals has a limit, then the sequence of the arithmetical means of these reals also converges and has the same limit. Simple counterexamples show that the converse is not true.
It should also be noticed that the Cesaro’s summability method inspired tauberian results in the general theory of series. Those results allow mathematicians to establish the convergence of a given series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ , if ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is summable by some method and satisfies certain additional constraints.^{9}
After loitering in the Devil’s toolshed, Abel suggested a technique now known as Abel’s summation method, or, sometimes, the AbelPoisson summation method. The Devil, it is well known, has work for idle hands.
Suppose that a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is given and, for a variable $x$ ranging over the open interval $\left(\mathrm{0,1}\right)$, consider the associated power series
${a}_{0}+{a}_{1}{x}^{1}+{a}_{2}{x}^{2}+\text{\u2026}+{a}_{n}{x}^{n}+\text{\u2026}$ .
It may happen that this series converges for each $r$ in $\left(\mathrm{0,1}\right)$, and therefore determines a concrete function $f\left(x\right)$ on the interval. It may happen that there exists a limit of $f\left(x\right)$ as $x$ tends to 1. Such is the series’ Abel sum. Cesaro’s method is compatible with the usual notion of the sum of a convergent series. So is Abel’s method. If a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ converges to $s$, then, by virtue of Abel’s classical theorem,^{10} the Abel sum of ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is also equal to $s$. The converse fails.
The Devil’s checks are never blank.
Example 13.
For any strictly positive integer $k$, the Abel method implies that the divergent series
$1{k}^{1}+{k}^{2}{k}^{3}+\text{\u2026}+{\left(1\right)}^{n1}{k}^{n1}+\text{\u2026}$ ,
sums to $1/(k+1)$, and, in particular, the Grandi’s series
$11+1\text{\u2026}+{\left(1\right)}^{n1}+\text{\u2026}$
sums to 1/2. The Grandi’s series is also summable by Cesaro’s method and its Cesaro’s sum is again equal to 1/2. This is not simply a happy coincidence. Abel’s method is substantially stronger than Cesaro’s. On the other hand, for a natural number $k$ strictly greater than 1, the series
$1{k}^{1}+{k}^{2}{k}^{3}+\text{\u2026}+{\left(1\right)}^{n1}{k}^{n1}+\text{\u2026}$ ,
is not Cesaro summable.
There are many other examples of series that are summable by Abel’s method and not by Cesaro’s method.
Example 14.
Consider the series $12+34+\text{\u2026}+{\left(1\right)}^{n1}n+\text{\u2026}$ which is trivially divergent and not Cesaro summable. Abel’s method works, but Cesaro’s method lapses. Indeed, if $0<x<1$, then the series $12{x}^{1}+3{x}^{2}4{x}^{3}+\text{\u2026}$ converges, and its ordinary sum is equal to $1/{(x+1)}^{2}$. If $x=1$, Abel’s method promptly yields the equality $12+34+{\left(1\right)}^{n1}n+\text{\u2026}=1/4$.
Abel’s method is very helpful in dealing with the products of infinite series. What follows represents one formula, but two series:
${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ ,
${b}_{0}+{b}_{1}+\text{\u2026}+{b}_{n}+\text{\u2026}$ .
Their product ${c}_{0}+{c}_{1}+\text{\u2026}+{c}_{n}+\text{\u2026}$ is defined by the equality
${c}_{n}={a}_{0}{b}_{n}+{a}_{1}{b}_{n1}+{a}_{2}{b}_{n2}+\text{\u2026}+{a}_{n}{b}_{0}$ $(n\in \mathbb{N})$
The product of two conditionally convergent series is not, in general, convergent. But if both series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ and ${b}_{0}+{b}_{1}+\text{\u2026}+{b}_{n}+\text{\u2026}$ are Abel summable, then their product ${c}_{0}+{c}_{1}+\text{\u2026}+{c}_{n}+\text{\u2026}$ is also Abel summable, and equal to the product of the Abel sums of ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ and ${b}_{0}+{b}_{1}+\text{\u2026}+{b}_{n}+\text{\u2026}$ .
This is a wonderfully elegant result, and, perhaps, the simplest result accessible to intuition that suggests that the techniques contrived by definition have a rich internal structure.
Example 15.
Using the fact that the sum of products is the product of sums—above, just above—there is, at once, the equality
$\left(11+11+\text{\u2026}\right)\left(11+11+\text{\u2026}\right)$
$=12+34+\text{\u2026}$ .
Example 16.
Emil Borel created, or discovered, several important summation methods. If
${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ ,
is a series trotting off into the unknown, then the function
$f\left(x\right)={e}^{x}\left(\mathrm{\Sigma}\left\{{s}_{n}{x}^{n}/n!:n\in \mathbb{N}\right\}\right)$ ,
makes sense—perfect sense, in fact—applied to the set of real numbers $x$, for which this function is well defined. It may happen that $f\left(x\right)$ has a finite limit $y$ as $x$ tends to +∞. If so, $y$ is called Borel’s sum of the series. If a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is convergent, then it is Borel summable, too, and its usual sum coincides with its Borel sum.
The proof is easy.
What follows involves the same Borel, but a different method: Borel integral summation.
Example 17.
The series $0!1!+2!3!+\dots +{(1)}^{n}n!+\text{\u2026}$ has as its Borel sum
$\underset{0}{\overset{\infty}{\int}}\frac{{e}^{t}}{1+t}dt$.^{11}
Example 18.
Consider the partial Riemann zeta function $\zeta \left(x\right)$ and the partial Dirichlet’s eta function $\eta \left(x\right)$ for all real numbers $x$ strictly greater than 1. The convergence of
$1/{1}^{x}+1/{2}^{x}+1/{3}^{x}+\text{\u2026}+1/{n}^{x}+\text{\u2026}$ ,
$1/{1}^{x}1/{2}^{x}+1/{3}^{x}\text{\u2026}+{\left(1\right)}^{n1}/{n}^{x}+\text{\u2026}$ ,
readily implies:
$\eta \left(x\right)=(1{2}^{\left(1x\right)})\zeta \left(x\right)$.
Even better, one can consider $1/{1}^{x}1/{2}^{x}+1/{3}^{x}\text{\u2026}+{\left(1\right)}^{n1}/{n}^{x}+\text{\u2026}$ for complex numbers $z$. Replacing $x$ with $z$, one obtains $1/{1}^{z}1/{2}^{z}+1/{3}^{z}\text{\u2026}+{\left(1\right)}^{n1}/{n}^{z}+\text{\u2026}$, which converges to a complex value whenever the real part of $z$ is strictly greater than 1. That $\eta \left(x\right)$ admits an analytic continuation on the whole complex plane C is a provable fact of life. On top of this, there is
$\eta \left(z\right)=(1{2}^{\left(1z\right)})\zeta \left(z\right)$,
where $\zeta \left(z\right)$ is an analytic function on the open set $\mathit{C}\mathit{}\backslash \left\{1\right\}$, extending $\zeta \left(x\right)$. By remembering that $\eta \left(1\right)=1/4$, one is finally able to discern that, against every expectation,
$1+2+3+\text{\u2026}+n+\text{\u2026}=\zeta \left(1\right)=1/12$.
This is an extraordinary result in mathematics, but it is also an extraordinary result in mathematics that has nontrivial applications in contemporary theoretical physics.
Did the inspired Euler, blind since middle age, who saw so much, see this?
The Theory of Infinite Summation
Suppose that a sequence $\left\{{g}_{n}:n\in \mathbb{N}\right\}$ of functions is given on some subset D of $\mathbb{R}$ such that there exists an accumulation point $d$ of D. Suppose also that, for each $n\in \mathbb{N}$, there exists a limit of ${g}_{n}\left(t\right)$ as $t\in \mathrm{D}$ tends to $d$, and that this limit is equal to 1.
Consider a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ and let the function series
$g\left(t\right)={a}_{0}{g}_{0}\left(t\right)+{a}_{1}{g}_{1}\left(t\right)+\text{\u2026}+{a}_{n}{g}_{n}\left(t\right)+\text{\u2026}$ $(t\in \mathrm{D})$
be its natural associate.
It may happen that the function $g\left(t\right)$ is welldefined on some neighbourhood of the point $d$ and, in addition, that there exists a limit $s=s\left({a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}\right)$ of $g\left(t\right)$ as $t\in \mathrm{D}$ tends to $d$. The series ${\mathrm{S}=a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ is summable, and its generalized sum is equal to $s$.
Example 19.
Assume that a sequence of functions $\left\{{g}_{n}:n\in \mathbb{N}\right\}$ satisfies these two conditions:
 all functions from $\left\{{g}_{n}:n\in \mathbb{N}\right\}$ are uniformly bounded; and
 for each $t\in \mathrm{D}$, the sequence $\left\{{g}_{n}\left(t\right):n\in \mathbb{N}\right\}$ is monotone.
If S converges to $s$, then S has $s$ as its sum.
Many classical summation methods correspond to the appropriate choice of a sequence of functions satisfying 1) and 2). For Cesaro’s summation method, the functions ${g}_{n}\left(n\in \mathbb{N}\right)$ can be defined on $\mathbb{N}$ in the following way:
${g}_{n}\left(m\right)=\left(m+1n\right)/\left(m+1\right)$ if $n\le m$,
and ${g}_{n}\left(m\right)=0$ if $n>m$
For Abel’s summation method, the functions ${g}_{n}\left(n\in \mathbb{N}\right)$ can be defined on the open interval (0,1)
${g}_{n}\left(t\right)={t}^{n}$ $\left(0<t<1\right)$
Taking the positive ray (0,+∞) of $\mathbb{R}$ as D, and defining the functions ${g}_{n}\left(n\in \mathbb{N}\right)$ by the formula
${g}_{n}\left(t\right)=1\left(\sum \left\{{t}^{k}/k!:k<n\right\}\right)/{e}^{t}$ $\left(t>0\right)$
one gets a certain version of Borel’s summation method.
There are other general schemes.^{12} Not everything is smooth sailing. Two different schemes may prove inconsistent in the sense that, for some concrete series, they yield two distinct generalized sums.
Example 20.
Quite often a summability scheme is formulated in terms of a realvalued matrix T with countably many rows and columns. Tsummability methods transform one infinite sequence into another in the expectation, or hope, that the transformation will yield a sequence better behaved so far as convergence goes. The celebrated Silverman–Toeplitz theorem establishes the necessary and sufficient conditions under which the class of all convergent sequences is transformed by T into itself.^{13}
Among the many interesting and important statements about matrix summability, I wish to mention one result.^{14} For a given matrix T and a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ , consider all those rearrangements of the terms of ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ that yield Tsummable series, and denote by the symbol $S({a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026 , T)}$ the set of all Tsums obtained in this way. It turns out that, changing ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ and T in all possible ways, the family of all sets $S({a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026 , T)}$ yields the family of all analytic, or Suslin, subsets of $\mathbb{R}$. This family is much wider than the family of all Borel subsets of $\mathbb{R}$.^{15}
Various summability schemes lead naturally to a large family of point sets in $\mathbb{R}$, and are connected with more or less delicate facts of classical descriptive set theory. Connections of this sort even touch upon the problem of the Lebesgue measurability of all subsets of $\mathbb{R}$.
From a more general point of view, any summability method may be described as a functional $f$ defined on some vector subspace L = dom($f$) of the space ${\mathbb{R}}^{\mathrm{N}}$ of all infinite real sequences. The elements of L are of the form $\left\{{s}_{n}:n\in \mathbb{N}\right\}$, where ${s}_{n}(n\in \mathbb{N})$ are partial finite sums of a series. As a rule, summability methods satisfy both linearity and regularity conditions. Linearity means that $f$ is a linear functional on L; regularity, that $f$ extends the standard linear functional limit defined on the space $c$ of all convergent sequences in ${\mathbb{R}}^{\mathrm{N}}$.
Sometimes, a third condition is imposed, a certain analogue of translation invariance. If $s$ is a sum of a series ${a}_{0}+{a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ and $r$ is a sum of ${a}_{1}+\text{\u2026}+{a}_{n}+\text{\u2026}$ , then $s={a}_{0}+r$.
Example 21.
No summability method satisfying the translation condition can be applied to the series $1+1+1+\text{\u2026}+1+\text{\u2026}$ . The proof is obvious: $s=1+\left(1+1+1+\text{\u2026}+1+\text{\u2026}\right)=1+s$, which implies that $0=1$.
Now, if a sequence $\left\{{t}_{n}:n\in \mathbb{N}\right\}$ does not belong to L and $r$ is an arbitrary real number, the functional $f$ can be extended to a linear functional $g$ defined on a larger vector subspace $K$ of ${\mathbb{R}}^{\mathrm{N}}$ such that
$\left\{{t}_{n}:n\in \mathbb{N}\right\}\in K$, $g\left(\left\{{t}_{n}:n\in \mathbb{N}\right\}\right)=r$.
This statement admits a constructive proof, one that does not rely on the axiom of choice.
Euler Vindicated
Euler’s wonderful intuition now stands vindicated:
For any concrete divergent series, there exists a linear and regular summability method that assigns a real number to the series as its sum.
A reservation follows at once. There exists no linear summability method satisfying the translation condition that assigns a finite sum to $1+2+3+\text{\u2026}+n+\text{\u2026}$ .
Suppose that
$$1+2+3+\text{\u2026}+n+\text{\u2026}=s,$$
where $s$ is a real number. Then:
$s1=2+3+\text{\u2026}+n+\text{\u2026}$ , and
$1=\left(s1\right)s=\left(21\right)+\left(32\right)+\text{\u2026}+\left(n+1n\right)+\text{\u2026}$
$=1+1+1+\text{\u2026}+1+\text{\u2026}$ ,
which is impossible.
If one wishes to assign certain generalized sums to all divergent series all at once, then one must appeal to strong forms of the axiom of choice. A standard application of Zorn’s lemma, for example, shows that that there exists a linear regular summation method whose domain is the whole of ${\mathbb{R}}^{\mathrm{N}}$. This result is useless in practice because of its extremely nonconstructive character. On the other hand, it follows from a fundamental result by Robert Solovay that, under countable forms of the axiom of choice, it is logically consistent to assume that the domain of any linear regular summability method is a first category subspace of ${\mathbb{R}}^{\mathrm{N}}$.^{16} Since ${\mathbb{R}}^{\mathrm{N}}$ is a Polish topological space, ${\mathbb{R}}^{\mathrm{N}}$ is of second category on itself. Consequently, the domain of any constructively defined linear and regular summability scheme is a very small part of ${\mathbb{R}}^{\mathrm{N}}$. Within the same weak fragment of set theory, one may assume that the class of all linear regular summation methods, equipped with its natural partial ordering, does not possess maximal elements.
Using uncountable forms of the axiom of choice, mathematicians were able to construct some remarkable linear functionals on various vector subspaces of ${\mathbb{R}}^{\mathrm{N}}$. Among them the best known is the Banach functional limit, or Lim, defined on the space of all bounded real sequences, Lim is linear, extends the standard functional limit, and is translation invariant.^{17} But the existence of such functionals always implies the existence of ultimately pathological subsets of $\mathbb{R}$, namely, sets nonmeasurable in the Lebesgue sense or sets not having the socalled Baire property.^{18}
Example 22.
Consider the following function series FS:
$\mathrm{cos}\left(t\right)+\mathrm{cos}\left(2t\right)+\mathrm{cos}\left(4t\right)+\text{\u2026}+\mathrm{cos}\left({2}^{n}t\right)+\text{\u2026 ,}$
where a variable $t$ ranges over $\mathbb{R}$. Assume that for each $t$ from a Lebesgue measurable subset of $\mathbb{R}$ with strictly positive measure, that FS sums to $f\left(t\right)$ by some linear, regular summation scheme satisfying the translation condition. It follows that $f$ is nonmeasurable in the Lebesgue sense. Such is Kolmogorov’s theorem.^{19}
Solovay’s profound result shows that there is no hope of effectively constructing a nonLebesgue measurable subset of $\mathbb{R}$ or,^{20} equivalently, a nonLebesgue measurable function acting from $\mathbb{R}$ into $\mathbb{R}$.^{21}
 “Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune demonstration.” Niels Henrik Abel, letter to Holmboe, January 1826. ↩
 See, for example, Tom Apostol, Mathematical Analysis (New York: Addison Wesley Longman, 1974); Jean Dieudonné, The Foundations of Modern Analysis (New York: Academic Press, 1960); Walter Rudin, Principles of Mathematical Analysis (New York: McGrawHill, 1976); Kenneth Ross, Elementary Analysis: The Theory of Calculus (New York: Springer, 1980); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 Compare Tom Apostol, Mathematical Analysis (New York: Addison Wesley Longman, 1974); Jean Dieudonné, The Foundations of Modern Analysis (New York: Academic Press, 1960); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Walter Rudin, Principles of Mathematical Analysis (New York: McGrawHill, 1976); Kenneth Ross, Elementary Analysis: The Theory of Calculus (New York: Springer, 1980); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 See Jean Dieudonné, The Foundations of Modern Analysis (New York: Academic Press, 1960). ↩
 See Tom Apostol, Mathematical Analysis (New York: Addison Wesley Longman, 1974); Gregory M. Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990); Kenneth Ross, Elementary Analysis: The Theory of Calculus (New York: Springer, 1980); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 See, for example, Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 See, for example, Tom Apostol, Mathematical Analysis (New York: Addison Wesley Longman, 1974); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Kenneth Ross, Elementary Analysis: The Theory of Calculus (New York: Springer, 1980); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 See Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Godfrey Hardy, Divergent Series (Oxford: Clarendon Press, 1949); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990); Louis Silverman, “On the Definition of the Sum of a Divergent Series,” University of Missouri Studies, Math. Series 1 (1913): 1–96; Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 For more details, see Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Godfrey Hardy, Divergent Series (Oxford: Clarendon Press, 1949); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 See, for example, Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990). ↩
 For details, see for example Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000); Godfrey Hardy, Divergent Series (Oxford: Clarendon Press, 1949); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990); Edmund Whittaker and George Watson, A Course of Modern Analysis, v. 1–v. 2 (Cambridge: Cambridge University Press, 1990). ↩
 See, for example, Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Godfrey Hardy, Divergent Series (Oxford: Clarendon Press, 1949); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990); Antoni Zygmund, Trigonometric Series, v. 1 (Cambridge: Cambridge University Press, 2002). ↩
 For more details, see Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000); Gregory Fichtenholz, Course in Differential and Integral Calculus, v. 2 (Moscow: Izdat. Nauka, 1966) (in Russian); Godfrey Hardy, Divergent Series (Oxford: Clarendon Press, 1949); Konrad Knopp, Theory and Application of Infinite Series (New York: Dover Publications, 1990); Louis Silverman, “On the Definition of the Sum of a Divergent Series,” University of Missouri Studies, Math. Series 1, (1913): 1–96; Antoni Zygmund, Trigonometric Series, v. 1 (Cambridge: Cambridge University Press, 2002). ↩
 George Lorentz and Karl Zeller, “Series Rearrangements and Analytic Sets,” Acta Mathematica 100, (1958): 149–169. ↩
 The theory of Borel and analytic sets in Polish topological spaces is presented in detail in the classical monographs of Kazimierz Kuratowski, Topology, vol. 1 (London–New York: Academic Press, 1966) and Nicolas Lusin, Leçons sur les Ensembles Analytiques et leurs Applications (Paris: GauthierVillars, 1930). See also Alexander Kechris, Classical Descriptive Set Theory (Berlin: Springer, 1995). ↩
 Robert M. Solovay, “A Model of Set Theory in Which Every Set of Reals is Lebesgue Measurable,” The Annals of Mathematics 2, no. 92 (1970): 1–56. ↩
 See, for example, Stan Wagon, The Banach–Tarski Paradox (Cambridge: Cambridge University Press, 1985). ↩
 See again Stan Wagon, The Banach–Tarski Paradox (Cambridge: Cambridge University Press, 1985). A lot of interesting and deep analogies between the Lebesgue measurability and Baire property of point sets are thoroughly discussed in a small but very informative textbook: John Oxtoby, Measure and Category (New York: Springer, 1971). ↩
 For more information about this and similar results, see Chapter 5 in Antoni Zygmund, Trigonometric Series, v. 1 (Cambridge: Cambridge University Press, 2002) or the extensive article: Yuri Lyubich, “Axiomatic Theory of Divergent Series and Cohomological Equations,” Fundamenta Mathematicae 198, no. 3, (2008): 263282. ↩
 Robert Solovay, “A Model of Set Theory in Which Every Set of Reals is Lebesgue Measurable,” The Annals of Mathematics 2, no. 92 (1970): 1–56. ↩
 Some other aspects of summability methods, treated from the point of view of modern functional analysis and the general theory of linear operators, are presented, for example, in Johann Boos, Classical and Modern Methods in Summability (Oxford: Clarendon Press, 2000) and Nelson Dunford and Jacob Schwartz, Linear Operators (General Theory), Part 1 (New York: WileyInterscience, 1988). ↩
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