Editing Series (mathematics) (section)

===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]].