Editing Series (mathematics) (section)

==History of the theory of infinite series==

===Development of infinite series===
Infinite series play an important role in modern analysis of [[Ancient Greece|Ancient Greek]] [[philosophy of motion]], particularly in [[Zeno's paradox|Zeno's paradoxes]].<ref name=":12">{{Citation |last=Huggett |first=Nick |title=Zeno's Paradoxes |year=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/spr2024/entries/paradox-zeno/ |access-date=2024-03-25 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref> The paradox of [[Achilles and the tortoise]] demonstrates that continuous motion would require an [[actual infinity]] of temporal instants, which was arguably an [[absurdity]]: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. [[Zeno of Elea|Zeno]] is said to have argued that therefore Achilles could ''never'' reach the tortoise, and thus that continuous movement must be an illusion. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the purely mathematical and imaginative side of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. However, in modern philosophy of motion the physical side of the problem remains open, with both philosophers and physicists doubting, like Zeno, that spatial motions are infinitely divisible: hypothetical reconciliations of [[quantum mechanics]] and [[general relativity]] in theories of [[quantum gravity]] often introduce [[Quantization (physics)|quantizations]] of [[spacetime]] at the [[Planck scale]].<ref>{{citation |last=Snyder |first=H. |title=Quantized space-time |journal=Physical Review |volume=67 |issue=1 |pages=38–41 |year=1947 |bibcode=1947PhRv...71...38S |doi=10.1103/PhysRev.71.38}}.</ref><ref>{{Cite web |date=2024-09-25 |title=The Unraveling of Space-Time |url=https://www.quantamagazine.org/the-unraveling-of-space-time-20240925/ |access-date=2024-10-11 |website=Quanta Magazine |language=en}}</ref>

[[Greek mathematics|Greek]] mathematician [[Archimedes]] produced the first known summation of an infinite series with a
method that is still used in the area of calculus today. He used the [[method of exhaustion]] to calculate the [[area]] under the arc of a [[parabola]] with the summation of an infinite series,<ref name=":6" /> and gave a remarkably accurate approximation of [[Pi|π]].<ref>{{cite web | title = A history of calculus |author1=O'Connor, J.J.  |author2=Robertson, E.F.  |name-list-style=amp | publisher = [[University of St Andrews]]| url = http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html |year=1996 |access-date= 2007-08-07}}</ref><ref>{{cite journal |title=Archimedes and Pi-Revisited. |last=Bidwell |first=James K. |date=30 November 1993 |journal=School Science and Mathematics |volume=94 |issue=3 |pages=127–129 |doi=10.1111/j.1949-8594.1994.tb15638.x }}</ref>

Mathematicians from the [[Kerala school of astronomy and mathematics|Kerala school]] were studying infinite series {{circa|1350 CE}}.<ref>{{cite web|url=http://www.manchester.ac.uk/discover/news/article/?id=2962|title=Indians predated Newton 'discovery' by 250 years|website=manchester.ac.uk}}</ref>

In the 17th century, [[James Gregory (astronomer and mathematician)|James Gregory]] worked in the new [[decimal]] system on infinite series and published several [[Maclaurin series]]. In 1715, a general method for constructing the [[Taylor series]] for all functions for which they exist was provided by [[Brook Taylor]]. [[Leonhard Euler]] in the 18th century, developed the theory of [[hypergeometric series]] and [[q-series]].

===Convergence criteria===
The investigation of the validity of infinite series is considered to begin with [[Carl Friedrich Gauss|Gauss]] in the 19th century. Euler had already considered the hypergeometric series

<math display=block>1 + \frac{\alpha\beta}{1\cdot\gamma}x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot \gamma(\gamma+1)}x^2 + \cdots</math>

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.

[[Cauchy]] (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by [[James Gregory (astronomer and mathematician)|Gregory]] (1668). [[Leonhard Euler]] and [[Carl Friedrich Gauss|Gauss]] had given various criteria, and [[Colin Maclaurin]] had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of [[power series]] by his expansion of a complex [[function (mathematics)|function]] in such a form.

[[Niels Henrik Abel|Abel]] (1826) in his memoir on the [[binomial series]]

<math display=block>1 + \frac{m}{1!}x + \frac{m(m-1)}{2!}x^2 + \cdots</math>

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of <math>m</math> and <math>x</math>. He showed the necessity of considering the subject of continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, and
the same may be said of [[Joseph Ludwig Raabe|Raabe]] (1832), who made the first elaborate investigation of the subject, of [[Augustus De Morgan|De Morgan]] (from 1842), whose
logarithmic test [[Paul du Bois-Reymond|DuBois-Reymond]] (1873) and [[Alfred Pringsheim|Pringsheim]] (1889) have
shown to fail within a certain region; of [[Joseph Louis François Bertrand|Bertrand]] (1842), [[Pierre Ossian Bonnet|Bonnet]]
(1843), [[Carl Johan Malmsten|Malmsten]] (1846, 1847, the latter without integration); [[George Gabriel Stokes|Stokes]] (1847), [[Paucker]] (1852), [[Chebyshev]] (1852), and [[Arndt]]
(1853).

General criteria began with [[Ernst Kummer|Kummer]] (1835), and have been studied by [[Gotthold Eisenstein|Eisenstein]] (1847), [[Weierstrass]] in his various
contributions to the theory of functions, [[Ulisse Dini|Dini]] (1867),
DuBois-Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete general theory.

===Uniform convergence===
The theory of [[uniform convergence]] was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it
successfully were [[Philipp Ludwig von Seidel|Seidel]] and [[George Gabriel Stokes|Stokes]] (1847–48). Cauchy took up the
problem again (1853), acknowledging Abel's criticism, and reaching
the same conclusions which Stokes had already found. Thomae used the
doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform
convergence, in spite of the demands of the theory of functions.

===Semi-convergence===
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not [[absolute convergence|absolutely convergent]].

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by [[Carl Johan Malmsten|Malmsten]] (1847). [[Schlömilch]] (''Zeitschrift'', Vol.I, p.&nbsp;192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and [[Faulhaber's formula|Bernoulli's function]]

<math display=block>F(x) = 1^n + 2^n + \cdots + (x - 1)^n.</math>

[[Angelo Genocchi|Genocchi]] (1852) has further contributed to the theory.

Among the early writers was [[Josef Hoene-Wronski|Wronski]], whose "loi suprême" (1815) was hardly recognized until [[Arthur Cayley|Cayley]] (1873) brought it into
prominence.

===Fourier series===
[[Fourier series]] were being investigated
as the result of physical considerations at the same time that
Gauss, Abel, and Cauchy were working out the theory of infinite
series. Series for the expansion of sines and cosines, of multiple
arcs in powers of the sine and cosine of the arc had been treated by
[[Jacob Bernoulli]] (1702) and his brother [[Johann Bernoulli]] (1701) and still
earlier by [[Franciscus Vieta|Vieta]]. Euler and [[Joseph Louis Lagrange|Lagrange]] simplified the subject,
as did [[Louis Poinsot|Poinsot]], [[Karl Schröter|Schröter]], [[James Whitbread Lee Glaisher|Glaisher]], and [[Ernst Kummer|Kummer]].

Fourier (1807) set for himself a different problem, to
expand a given function of {{tmath|x}} in terms of the sines or cosines of
multiples of {{tmath|x}}, a problem which he embodied in his ''[[Théorie analytique de la chaleur]]'' (1822). Euler had already given the formulas for determining the coefficients in the series;
Fourier was the first to assert and attempt to prove the general
theorem. [[Siméon Denis Poisson|Poisson]] (1820–23) also attacked the problem from a
different standpoint. Fourier did not, however, settle the question
of convergence of his series, a matter left for [[Augustin Louis Cauchy|Cauchy]] (1826) to
attempt and for Dirichlet (1829) to handle in a thoroughly
scientific manner (see [[convergence of Fourier series]]). Dirichlet's treatment (''[[Journal für die reine und angewandte Mathematik|Crelle]]'', 1829), of trigonometric series was the subject of criticism and improvement by
Riemann (1854), Heine, [[Rudolf Lipschitz|Lipschitz]], [[Ludwig Schläfli|Schläfli]], and
[[Paul du Bois-Reymond|du Bois-Reymond]]. Among other prominent contributors to the theory of
trigonometric and Fourier series were [[Ulisse Dini|Dini]], [[Charles Hermite|Hermite]], [[Georges Henri Halphen|Halphen]],
Krause, Byerly and [[Paul Émile Appell|Appell]].