Vizualizare sursă pentru „Utilizator:Aether/Groapa cu nisip”

[[Image:Pm1234 Ground.png|thumb|Primele mii de sume parţiale 1 − 2 + 3 − 4 + …]]

În [[matematică]], '''1 - 2 + 3 - 4 + …''' este [[Serie_(matematică)#C.C3.A2teva_tipuri_de_serii_infinite|seria infinită]] a cărei termeni sunt [[Număr întreg|numere întregi]] [[număr pozitiv|pozitive]] succesive, cu semne variabile. Folosind sigma ca notaţie a însumării, suma primilor ''m'' termeni ai seriei pot fi exprimaţi ca:

:<math alt="Summation from n equals 1 to m of the series n * (-1)^(n-1)">\sum_{n=1}^m n(-1)^{n-1}.</math>

Seria infinită [[serie divergentă|se diverge]], adică secvenţa sa de [[sumă parţială|sume parţiale]], {{nowrap|(1, −1, 2, −2, …)}}, nu tinde înspre o [[limita unei secvenţe|limită]] finită. Cu toate acestea, în mijlocul secolului al XVIII-lea, [[Leonhard Euler]] a redactat despre ceea ce, în opinia lui, era o ecuaţie paradoxală:

:<math alt="1-2+3-4+...=1/4">1-2+3-4+\cdots=\frac{1}{4}.</math>

O explicaţie riguroasă a acestei egalări avea să apară mult mai târziu. Începând în 1890, [[Ernesto Cesàro]], [[Émile Borel]] şi alţii au investigat metode bine definite (pe baza [[axiomă|axiomelor]]) pentru a rezulta în sume generalizate pentru serii divergente, printre care şi noi interpretări a încercărilor lui Euler. Multe dintre metodele de sumare alocă pentru {{nowrap|1 − 2 + 3 − 4 + …}} „suma” de {{Fracţie|1|4}} într-un final. [[Sumaţia lui Cesàro]] este una din puţinele procedee care nu însumează  {{nowrap|1 − 2 + 3 − 4 + …}}, aşadar seria este un exemplu unde trebuie folosită o metodă mai eficientă, precum [[serii divergente|sumaţia lui Abel]].

Seria 1 − 2 + 3 − 4 + … este strâns legată de [[seria lui Grandi]], {{nowrap|1 − 1 + 1 − 1 + …}}. Euler le-a tratat pe acestea ca fiind cazuri speciale a {{nowrap|1 − 2<sup>''n''</sup> + 3<sup>''n''</sup> − 4<sup>''n''</sup> + …}} pentru ''n'' arbitrar, o gamă de cercetare extinzând activitatea sa asupra [[problema lui Basel|problemei lui Basel]] şi călăuzindu-l spre [[ecuaţie funcţională|ecuaţia funcţională]] a ceea ce este cunoscut azi ca [[funcţia eta a lui Dirichlet]] şi [[funcţia zeta a lui Riemann]].

==Divergenţă==
Termenii seriei nu se apropie de [[0 (cifră)|0]]; prin urmare, {{nowrap|1 − 2 + 3 − 4 + …}} se diverge, conform [[testul de divergenţă|testului de termeni]]. Ca bază pentru analiză în secţiunile ulterioare, este de asemenea util să se observe divergenţa la nivel fundamental. Prin definiţie, convergenţa sau divergenţa unei serii infinite este determinată de [[limita unei secvenţe|convergenţa sau divergenţa]] secvenţei sale de sume parţiale, iar în acest caz, sumele parţiale pentru {{nowrap|1 − 2 + 3 − 4 + …}} sunt:<ref>Hardy p.8</ref>
:1 = '''1''',
:1 − 2 = '''−1''',
:1 − 2 + 3 = '''2''',
:1 − 2 + 3 − 4 = '''−2''',
:1 − 2 + 3 − 4 + 5 = '''3''',
:1 − 2 + 3 − 4 + 5 − 6 = '''−3''',
:…
Această succesiune este notabilă pentru abordarea fiecărui [[număr întreg]] - chiar şi 0 dacă s-ar lua în calcul suma parţială vidă - şi, astfel, stabileşte numerabilitatea setului <math>\mathbb{Z}</math> de întregi. <ref>Beals p.23</ref>. Secvenţa de sume parţiale arată clar că seria nu se converge spre un număr particular (pentru orice limită ''x'' propusă, se poate găsi un punct dincolo de aceasta, pentru care toate sumele parţiale ulterioare nu sunt incluse în intervalul [''x''-1, ''x''+1]), deci {{nowrap|1 − 2 + 3 − 4 + …}} se diverge.

==Euristică pentru însumare==
===Stabilitate şi linearitate===
Din moment ce termenii 1, −2, 3, −4, 5, −6 ş.a.m.d. urmează un tipar simplu, seria {{nowrap|1 − 2 + 3 − 4 + …}} poate fi manipulată mutarea poziţiei şi adunarea termen cu termen pentru a obţine o valoare numerică. Dacă are sens a afirma {{nowrap|1=''s'' = 1 − 2 + 3 − 4 + …}} pentru un număr ''s'' oarecare, următoarele manipulări demonstrează că {{nowrap|1=''s'' = {{fracţie|1|4}}:}}<ref>Hardy (p.6) prezintă derivarea în legătură cu evaluarea seriei lui Grandi, {{nowrap|1 − 1 + 1 − 1 + …}}.</ref> 

<math>
\begin{array}{rclllll}
4s&=& &(1-2+3-4+\cdots) & +(1-2+3-4+\cdots) & +(1-2+3-4+\cdots) &+(1-2+3-4+\cdots) \\
 &=& &(1-2+3-4+\cdots) & +1+(-2+3-4+5+\cdots) & +1+(-2+3-4+5+\cdots) &-1+(3-4+5-6\cdots) \\
 &=&1+[&(1-2-2+3) & +(-2+3+3-4) & +(3-4-4+5) &+(-4+5+5-6)+\cdots] \\
 &=&1+[&0+0+0+0+\cdots] \\
4s&=&1
\end{array}
</math>
[[Image:Pm1234 linearity.svg|thumb|right|Însumând 4 copii a {{nowrap|1 − 2 + 3 − 4 + …,}} folosind doar schimbarea poziţiei şi adunare termen cu termen, rezultatul reiese 1. Partea stângă şi partea dreaptă fiecare manifestă două copii a {{nowrap|1 − 2 + 3 − 4 + …}}, completând demonstrarea pentru {{nowrap|1 − 1 + 1 − 1 + ….}}]] Deci, <math>s=\frac{1}{4}</math>. Această derivare este ilustrată grafic în imaginea din dreapta.

Deşi 1 − 2 + 3 − 4 + … nu are un rezultat în sensul obişnuit, ecuaţia {{nowrap|1=''s'' = 1 − 2 + 3 − 4 + … = {{fracţie|1|4}}}} poate fi tolerată ca şi cel mai firesc răspuns, în caz că astfel de sumă trebuie definită. O definiţie generalizată a „sumei” unei serii divergente este metoda sumaţiei, sau a sumabilităţii, care adună o submulţime a tuturor seriilor posibile. Există mai multe metode diferite (unele fiind descrise [[#Metode specifice|mai jos]]) ce sunt caracterizate prin proprietăţile pe care le împart cu sumaţiile obişnuite. Ceea ce manipulările de mai sus demonstrează cu adevărat este: Fie orice metodă de sumabilitate care este [[Serii divergente|lineară şi stabilă]] şi însumează seria {{nowrap|1 − 2 + 3 − 4 + …}}, suma produsă este {{fracţie|1|4}}. Mai mult, întrucât

<math>
\begin{array}{rcllll}
2s & = & &(1-2+3-4+\cdots) & + & (1-2+3-4+\cdots) \\
 & = & 1 + &(-2+3-4+\cdots) & + 1 - 2 & + (3-4+5\cdots) \\
 & = & 0 + &(-2+3)+(3-4)+ (-4+5)+\cdots \\
2s & = & &1-1+1-1\cdots \\
\end{array}
</math>

asemenea metodă trebuie să însumeze şi [[seria lui Grandi]] ca {{nowrap|1=1 − 1 + 1 − 1 + … = {{fracţie|1|2}}.}}

===Produsul lui Cauchy===
In 1891, [[Ernesto Cesàro]] expressed hope that divergent series would be rigorously brought into [[calculus]], pointing out, "One already writes {{nowrap|1=(1 − 1 + 1 − 1 + …)<sup>2</sup> = 1 − 2 + 3 − 4 + …}} and asserts that both the sides are equal to {{frac|1|4}}."<ref>Ferraro, p.130.</ref> For Cesàro, this equation was an application of a theorem he had published the previous year, one that may be identified as the first theorem in the history of summable divergent series. The details on his summation method are [[#Cesàro and Hölder|below]]; the central idea is that {{nowrap|1 − 2 + 3 − 4 + …}} is the [[Cauchy product]] of {{nowrap|1 − 1 + 1 − 1 + …}} with {{nowrap|1 − 1 + 1 − 1 + …}}.


The Cauchy product of two infinite series is defined even when both of them are divergent. In the case where Σ''a''<sub>''n''</sub> = Σ''b''<sub>''n''</sub> = Σ(−1)<sup>''n''</sup>, the terms of the Cauchy product are given by the finite diagonal sums

:<math>\begin{array}{rcl}
c_n & = &\displaystyle \sum_{k=0}^n a_k b_{n-k}=\sum_{k=0}^n (-1)^k (-1)^{n-k} \\[1em]
 & = &\displaystyle \sum_{k=0}^n (-1)^n = (-1)^n(n+1).
\end{array}</math>

The product series is then
:<math>\sum_{n=0}^\infty(-1)^n(n+1) = 1-2+3-4+\cdots.</math>

Thus a summation method that respects the Cauchy product of two series and sums {{nowrap|1=1 − 1 + 1 − 1 + … = {{frac|1|2}},}} will also sum {{nowrap|1=1 − 2 + 3 − 4 + … = {{frac|1|4}}}}. With the result of the previous section, this implies an equivalence between summability of {{nowrap|1 − 1 + 1 − 1 + …}} and {{nowrap|1 − 2 + 3 − 4 + …}} with methods that are linear, stable, and respect the Cauchy product.

Cesàro's theorem is a subtle example. The series {{nowrap|1=1 − 1 + 1 − 1 + …}} is Cesàro-summable in the weakest sense, called {{nowrap|(C, 1)-summable,}} while {{nowrap|1=1 − 2 + 3 − 4 + …}} requires a stronger form of Cesàro's theorem<ref>Hardy, p.3; Weidlich, pp.52–55.</ref>, being {{nowrap|(C, 2)-summable.}} Since all forms of Cesàro's theorem are linear and stable, the values of the sums are as we have calculated.

==Metode specifice==
===Cesàro şi Hölder===
[[Image:Pm1234 means.svg|thumb|right|Data about the (H, 2) sum of {{frac|1|4}}]]

To find the (C, 1) [[Cesàro summation|Cesàro sum]] of 1 − 2 + 3 − 4 + …, if it exists, one needs to compute the [[arithmetic mean]]s of the partial sums of the series.
The partial sums are

:1, −1, 2, −2, 3, −3, …,

and the arithmetic means of these partial sums are

:1, 0, {{frac|2|3}}, 0, {{frac|3|5}}, 0, {{frac|4|7}}, ….

This sequence of means does not converge, so 1 − 2 + 3 − 4 + … is not Cesàro summable.

There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of (H, ''n'') methods for [[natural number]]s ''n''. The (H, 1) sum is Cesàro summation, and higher methods repeat the computation of means. Above, the even means converge to {{frac|1|2}}, while the odd means are all equal to 0, so the means ''of'' the means converge to the average of 0 and {{frac|1|2}}, namely {{frac|1|4}}.<ref>Hardy, p.9. For the full details of the calculation, see Weidlich, pp.17–18.</ref> So {{nowrap|1 − 2 + 3 − 4 + …}} is (H, 2) summable to {{frac|1|4}}.

The "H" stands for [[Otto Hölder]], who first proved in 1882 what mathematicians now think of as the connection between [[Divergent series#Abelian means|Abel summation]] and (H, ''n'') summation; {{nowrap|1 − 2 + 3 − 4 + …}} was his first example.<ref>Ferraro, p.118; Tucciarone, p.10. Ferraro criticizes Tucciarone's explanation (p.7) of how Hölder himself thought of the general result, but the two authors' explanations of Hölder's treatment of 1 − 2 + 3 − 4 + … are similar.</ref> The fact that {{frac|1|4}} is the (H, 2) sum of {{nowrap|1 − 2 + 3 − 4 + …}} guarantees that it is the Abel sum as well; this will also be proved directly below.

The other commonly formulated generalization of Cesàro summation is the sequence of (C, ''n'') methods. It has been proven that (C, ''n'') summation and (H, ''n'') summation always give the same results, but they have different historical backgrounds. In 1887, Cesàro came close to stating the definition of (C, ''n'') summation, but he gave only a few examples. In particular, he summed {{nowrap|1 − 2 + 3 − 4 + …,}} to {{frac|1|4}} by a method that may be rephrased as (C, ''n'') but was not justified as such at the time. He formally defined the (C, n) methods in 1890 in order to state his theorem that the Cauchy product of a (C, ''n'')-summable series and a (C, ''m'')-summable series is (C, ''m'' + ''n'' + 1)-summable.<ref>Ferraro, pp.123–128.</ref>

===Sumaţia lui Abel===
[[Image:Pm1234 Abel.svg|thumb|left|120px|Some partials of 1−2''x''+3''x''<sup>2</sup>+…; 1/(1 + ''x'')<sup>2</sup>; and limits at 1]]
In a 1749 report, [[Leonhard Euler]] admits that the series diverges but prepares to sum it anyway:

{{blockquote|1=…when it is said that the sum of this series 1−2+3−4+5−6 etc. is {{frac|1|4}}, that must appear paradoxical. For by adding 100 terms of this series, we get –50, however, the sum of 101 terms gives +51, which is quite different from {{frac|1|4}} and becomes still greater when one increases the number of terms. But I have already noticed at a previous time, that it is necessary to give to the word ''sum'' a more extended meaning….<ref>Euler et al., p.2. Although the paper was written in 1749, it was not published until 1768.</ref>}}

Euler proposed a generalization of the word "sum" several times; see ''[[Euler on infinite series]]''. In the case of {{nowrap|1 − 2 + 3 − 4 + …,}} his ideas are similar to what is now known as [[Divergent series#Abelian means|Abel summation]]:

{{blockquote|1=…it is no more doubtful that the sum of this series 1−2+3−4+5 + etc. is {{frac|1|4}}; since it arises from the expansion of the formula {{frac|1|(1+1)<sup>2</sup>}}, whose value is incontestably {{frac|1|4}}. The idea becomes clearer by considering the general series 1 − 2''x'' + 3''x''<sup>2</sup> − 4''x''<sup>3</sup> + 5''x''<sup>4</sup> − 6''x''<sup>5</sup> + ''&c.'' that arises while expanding the expression {{frac|1|(1+''x'')<sup>2</sup>}}, which this series is indeed equal to after we set ''x'' = 1.<ref>Euler et al., pp.3, 25.</ref>}}

There are many ways to see that, at least for [[absolute value]]s |''x''| < 1, Euler is right in that
:<math>1-2x+3x^2-4x^3+\cdots = \frac{1}{(1+x)^2}.</math>
One can take the [[Taylor series|Taylor expansion]] of the right-hand side, or apply the formal [[polynomial long division|long division process for polynomials]]. Starting from the left-hand side, one can follow the general heuristics above and try multiplying by (1+''x'') twice or [[square (algebra)|squaring]] the [[geometric series]] {{nowrap|1 − ''x'' + ''x''<sup>2</sup> − ….}} Euler also seems to suggest [[derivative|differentiating]] the latter series term by term.<ref>For example, Lavine (p.23) advocates long division but does not carry it out; Vretblad (p.231) calculates the Cauchy product. Euler's advice is vague; see Euler et al., pp.3, 26. [[John C. Baez|John Baez]] even suggests a category-theoretic method involving multiply [[pointed set]]s and the [[quantum harmonic oscillator]]. Baez, John C. [http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf Euler's Proof That 1 + 2 + 3 + … = 1/12 (PDF).] math.ucr.edu (December 19, 2003). Retrieved on March 11, 2007.</ref>

In the modern view, the series 1 − 2''x'' + 3''x''<sup>2</sup> − 4''x''<sup>3</sup> + … does not define a [[function (mathematics)|function]] at {{nowrap|1=''x'' = 1,}} so that value cannot simply be substituted into the resulting expression. Since the function is defined for all {{nowrap|&#124;''x''&#124; < 1,}} one can still take the limit as ''x'' approaches 1, and this is the definition of the Abel sum:

:<math>\lim_{x\rightarrow 1^{-}}\sum_{n=1}^\infty n(-x)^{n-1} = \lim_{x\rightarrow 1^{-}}\frac{1}{(1+x)^2} = \frac14.</math>

===Euler şi Borel===
[[Image:Pm1234 Euler.svg|right|thumb|Euler summation to {{frac|1|2}} − {{frac|1|4}}]]
Euler applied another technique to the series: the [[Binomial transform#Euler transform|Euler transform]], one of his own inventions. To compute the Euler transform, one begins with the sequence of positive terms that makes up the alternating series—in this case {{nowrap|1, 2, 3, 4, ….}} The first element of this sequence is labeled ''a''<sub>0</sub>.

Next one needs the sequence of [[difference operator|forward differences]] among {{nowrap|1, 2, 3, 4, …}}; this is just {{nowrap|1, 1, 1, 1, ….}} The first element of ''this'' sequence is labeled Δ''a''<sub>0</sub>. The Euler transform also depends on differences of differences, and higher [[iterated function|iterations]], but all the forward differences among {{nowrap|1, 1, 1, 1, …}} are 0. The Euler transform of {{nowrap|1 − 2 + 3 − 4 + …}} is then defined as 

:<math>\frac12 a_0-\frac14\Delta a_0 +\frac18\Delta^2 a_0 -\cdots = \frac12-\frac14.</math>

In modern terminology, one says that {{nowrap|1 − 2 + 3 − 4 + …}} is [[Euler summation|Euler summable]] to {{frac|1|4}}.

The Euler summability implies another kind of summability as well. Representing {{nowrap|1 − 2 + 3 − 4 + …}} as

:<math>\sum_{k=0}^\infty a_k = \sum_{k=0}^\infty(-1)^k(k+1),</math>

one has the related everywhere-convergent series

:<math>a(x) = \sum_{k=0}^\infty\frac{(-1)^k(k+1)x^k}{k!} = e^{-x}(1-x).</math>

The [[Borel summation|Borel sum]] of 1 − 2 + 3 − 4 + … is therefore<ref>Weidlich p. 59</ref>

:<math>\int_0^\infty e^{-x}a(x)\,dx = \int_0^\infty e^{-2x}(1-x)\,dx = \frac12-\frac14.</math>

===Separarea proporţiilor===
Saichev and Woyczyński arrive at {{nowrap|1=1 − 2 + 3 − 4 + … = {{frac|1|4}}}} by applying only two physical principles: ''infinitesimal relaxation'' and ''separation of scales''. To be precise, these principles lead them to define a broad family of "''φ''-summation methods", all of which sum the series to {{frac|1|4}}:

*If ''φ''(''x'') is a function whose first and second derivatives are continuous and integrable over (0, ∞), such that φ(0) = 1 and the limits of φ(''x'') and ''x''φ(''x'') at +∞ are both 0, then<ref>Saichev and Woyczyński, pp.260–264.</ref>

::<math>\lim_{\delta\rightarrow0}\sum_{m=0}^\infty (-1)^m(m+1)\varphi(\delta m) = \frac14.</math>

This result generalizes Abel summation, which is recovered by letting ''φ''(''x'') = exp(−''x''). The general statement can be proved by pairing up the terms in the series over ''m'' and converting the expression into a [[Riemann integral]]. For the latter step, the [[Summation of Grandi's series#Separation of scales|corresponding proof]] for {{nowrap|1 − 1 + 1 − 1 + …}} applies the [[mean value theorem]], but here one needs the stronger Lagrange form of [[Taylor's theorem]].

==Generalizări==
[[Image:Pm1234 Euler1755 I-V.png|thumb|Euler sums similar series in the 1755 ''Institutiones'']]
The threefold Cauchy product of {{nowrap|1 − 1 + 1 − 1 + …}} is {{nowrap|1 − 3 + 6 − 10 + …,}} the alternating series of [[triangular number]]s; its Abel and Euler sum is {{frac|1|8}}.<ref>Kline, p.313.</ref> The fourfold Cauchy product of {{nowrap|1 − 1 + 1 − 1 + …}} is {{nowrap|1 − 4 + 10 − 20 + …,}} the alternating series of [[tetrahedral number]]s, whose Abel sum is {{frac|1|16}}.

Another generalization of 1 − 2 + 3 − 4 + … in a slightly different direction is the series {{nowrap|1 − 2<sup>''n''</sup> + 3<sup>''n''</sup> − 4<sup>''n''</sup> + …}} for other values of ''n''. For positive integers ''n'', these series have the following Abel sums:<ref>Knopp, p.491; there appears to be an error at this point in Hardy, p.3.</ref>
:<math>1-2^{n}+3^{n}-\cdots = \frac{2^{n+1}-1}{n+1}B_{n+1}</math>
where ''B''<sub>''n''</sub> are the [[Bernoulli number]]s. For even ''n'', this reduces to

:<math>1-2^{2k}+3^{2k}-\cdots = 0.</math>

This last sum became an object of particular ridicule by [[Niels Henrik Abel]] in 1826:

:"Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them. One can get out of them what one wants if one uses them, and it is they which have made so much unhappiness and so many paradoxes. Can one think of anything more appalling than to say that

::0 = 1 − 2<sup>''n''</sup> + 3<sup>''n''</sup> − 4<sup>''n''</sup> + etc.

:where ''n'' is a positive number. Here's something to laugh at, friends."<ref>Grattan-Guinness, p.80. See Markushevich, p.48, for a different translation from the original [[French language|French]]; the tone remains the same.</ref>

Cesàro's teacher, [[Eugène Charles Catalan]], also disparaged divergent series. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for {{nowrap|1 − 2<sup>''n''</sup> + 3<sup>''n''</sup> − 4<sup>''n''</sup> + …}} as "absurd equalities", and in 1883 Cesàro expressed a typical view of the time that the formulas were false but still somehow formally useful. Finally, in his 1890 ''Sur la multiplication des séries'', Cesàro took a modern approach starting from definitions.<ref>Ferraro, pp.120–128.</ref>

The series are also studied for non-integer values of ''n''; these make up the [[Dirichlet eta function]]. Part of Euler's motivation for studying series related to {{nowrap|1 − 2 + 3 − 4 + …}} was the [[functional equation]] of the eta function, which leads directly to the functional equation of the [[Riemann zeta function]]. Euler had already become famous for finding the values of these functions at positive [[even and odd numbers|even integers]] (including the [[Basel problem]]), and he was attempting to find the values at the positive [[even and odd numbers|odd integers]] (including [[Apéry's constant]]) as well, a problem that remains elusive today. The eta function in particular is easier to deal with by Euler's methods because its [[Dirichlet series]] is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges.<ref>Euler et al., pp.20–25.</ref> For example, the counterpart of {{nowrap|1 − 2 + 3 − 4 + …}} in the zeta function is the non-alternating series {{nowrap|[[1 + 2 + 3 + 4 + · · ·|1 + 2 + 3 + 4 + …]]}}, which has deep applications in modern [[physics]] but requires much stronger methods to sum.

==Bibliografie==
;Generală
{{refbegin}}
*{{cite book |last=Beals |first=Richard |title=Analysis: an introduction |year=2004 |publisher=Cambridge UP |isbn= 0-521-60047-2}}
*{{cite book |last=Davis |first=Harry F. |title=Fourier Series and Orthogonal Functions |year=1989 |month=May |publisher=Dover |isbn= 0-486-65973-9}}
*{{cite web |author=Euler, Leonhard; Lucas Willis; and Thomas J Osler |title=Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series |year=2006 |publisher=The Euler Archive |url=http://www.math.dartmouth.edu/~euler/pages/E352.html |accessdate=2007-03-22}} Originally published as {{cite journal |last=Euler |first=Leonhard |title=Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques |journal=Memoires de l'academie des sciences de Berlin |year=1768 |volume=17 |pages=83–106}}
*{{cite journal |last=Ferraro |first=Giovanni |title=The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics |journal=Archive for History of Exact Sciences |year=1999 |month=June |volume=54 |issue=2 |pages=101–135 |doi=10.1007/s004070050036}}
*{{cite book |last=Grattan-Guinness |authorlink=Ivor Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn= 0-262-07034-0}}
*{{cite book |last=Hardy |first=G.H. |authorlink=G. H. Hardy |title=Divergent Series |year=1949 |publisher=Clarendon Press |id={{LCCN|91|0|75377}}}}
*{{cite journal |last=Kline |first=Morris |authorlink=Morris Kline |title=Euler and Infinite Series |journal=Mathematics Magazine |volume=56 |issue=5 |year=1983 |month=November |pages=307–314 |url=http://links.jstor.org/sici?sici=0025-570X%28198311%2956%3A5%3C307%3AEAIS%3E2.0.CO%3B2-M}}
*{{cite book |first=Shaughan |last=Lavine |title=Understanding the Infinite |year=1994 |publisher=Harvard UP |isbn= 0674920961}}
*{{cite book |last=Markushevich |first=A.I. |title=Series: fundamental concepts with historical exposition |year=1967 |edition=English translation of 3rd revised edition (1961) in Russian |publisher=Hindustan Pub. Corp. |id={{LCCN|68|0|17528}}}}
*{{cite book |author=Saichev, A.I., and W.A. Woyczyński |title=Distributions in the physical and engineering sciences, Volume 1 |publisher=Birkhaüser |year=1996 |isbn= 0-8176-3924-1}}
*{{cite journal |last=Tucciarone |first=John |title=The development of the theory of summable divergent series from 1880 to 1925 |journal=Archive for History of Exact Sciences |volume=10 |issue=1-2 |year=1973 |month=January |pages=1–40 |doi=10.1007/BF00343405}}
*{{cite book |first=Anders |last=Vretblad |title=Fourier Analysis and Its Applications |year=2003 |publisher=Springer |isbn= 0387008365}}
*{{cite book |last=Weidlich |first=John E. |title=Summability methods for divergent series |year=1950 |month=June |publisher=Stanford M.S. theses|id={{OCLC|38624384}}}}
{{refend}}

;Specifică
<references />