Why did we make a new name for infinite sum? Was something wrong with calling it an infinite sum, or is it highlighting a difference between finite and infinite?

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    $\begingroup$ Sums are associative, commutative, and don't take you out of an abelian group. Series are sensitive to reordering, may e.g. produce an irrational value from rational terms, or not converge at all $\endgroup$ – Hagen von Eitzen May 20 '16 at 10:30
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    $\begingroup$ A series is not an "infinite sum", it seems an infinite sum but it doesnt have many properties of sums... it is more like and "ordered sum". It represent the limit of a sequence (that is a kind of function). When a sum is infinite it lose many properties of the classical finite sums. $\endgroup$ – Masacroso May 20 '16 at 10:41
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    $\begingroup$ @barakmanos, 'series' can be singular as well (for example. TV-series) $\endgroup$ – Yuriy S May 20 '16 at 11:16
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    $\begingroup$ The word series normally refers to the formal expression with infinitely many numbers (terms) with plus symbols in between them, $a_1+a_2+a_3+\dots$. This "object" can be considered even if there does not exist a number which can be described as the sum of the series. So in a sense, a series is more like an infinite addition (which may or may not lead to a "result", or sum). $\endgroup$ – Jeppe Stig Nielsen May 20 '16 at 14:23
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    $\begingroup$ @barakmanos : "Series" is a Latin word. As it happens, the nominative singular form and the nominative plural form are identical. The final "s" doesn't necessarily indicate a plural (like "radius"). $\endgroup$ – MPW May 20 '16 at 17:26

To mention more specifically what some of the comments above are referring to: series do not have many of the properties of sums, so the name "infinite sum" would be misleading. A notorious example of this is that in a series, the value of the series can depend on the order in which we "sum" up the terms. There is a famous theorem called the Riemann rearrangement Theorem which says that if a series is conditionally convergent (i.e. the series converges but it does not converge absolutely) then one can permute the order of the terms of the series to make the series add up to any value you like. See for example http://individual.utoronto.ca/jordanbell/notes/summable.pdf for a proof of this.

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    $\begingroup$ I don't really believe "infinite sum" is not used because of the subtle pathologies one does not find in finite sums. (One could equally well say that "infinite series" is misleading because of the subtle pathologies one does not find in finite series.) I think an infinite series is certainly a mathematical precision of the concept of "an infinite sum," and just like precisions of any intuitive notion, there are subtle problems: e.g. "continuous function" makes precise the notion that the graph is an unbroken curve, but there are "broken curves" that are continuous functions. $\endgroup$ – Pete L. Clark May 20 '16 at 14:35
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    $\begingroup$ I suppose I think that the actual answer is the usual, boring one to "Why is X not called Y?" questions: namely, that X is called X is mostly historical accident, so if you choose Y to be what you now think is a good, sensible name, the chance that X got called Y is small. $\endgroup$ – Pete L. Clark May 20 '16 at 14:37
  • $\begingroup$ I think that is probably fair enough. Perhaps my answer helps explain to the OP why it is not a good idea to blur the concepts into one. $\endgroup$ – Josh R May 20 '16 at 16:02

See Earliest Known Uses of Some of the Words of Mathematics (Series):

SERIES. According to Smith (vol. 2, page 481), "The early writers often used proportio to designate a series, and this usage is found as late as the 18th century."

According to Smith (vol. 2, page 497), "The change to the name ’series' seems to have been due to writers of the 17th century. Even as late as the 1693 edition of his Algebra, however, Wallis used the expression 'infinite progression' for infinite series."

In the English translation of Wallis' Algebra (translated by him and published in 1685), Wallis wrote:

Now (to return where we left off:) Those Approximations (in the Arithmetick of Infinites) above mentioned, (for the Circle or Ellipse, and the Hyperbola;) have given occasion to others (as is before intimated,) to make further inquiry into that subject; and seek out other the like Approximations, (or continual approaches) in other cases. Which are now wont to be called by the name of Infinite Series, or Converging Series, or other names of like import.

Series comes from Latin series, from serere ‎(“to join together, bind”).


A series is a sum that results from adding up terms of a sequence (which has a well defined order). One should understand that mathematical concepts were viewed differently in the past - the notion of infinity, which we now take for granted, was not thrown around lightly, and notation also wasn't what it is now.

The same root as series (which is explained by @Mauro to come from Latin) is shared by all major romance languages I checked. In German, other Germanic languages, and most Slavic languages, the equivalent of the word order is used (and matches the word for order used for taxonomic rank in biology). In Slovenian, the word is vrsta, which loosely translates as queue/line-up (or possibly sort/species, but in this case the similarity with taxonimical classification is a coincidence).

  • $\begingroup$ Interesting then that one Germanic language prefers the romantic version. But given the polyglot nature of the English language, it is perhaps not surprising that English mathematicians chose to simply borrow the Latin name rather than translate it. $\endgroup$ – Paul Sinclair May 20 '16 at 16:00
  • $\begingroup$ English is known to adopt latinate forms, adjust the pronunciation and use it without further consideration. Also, romance and germanic influence are sort of 50/50 in english. So it's not surprising that this happened. Anyway, the etymology of the currently used form reveals the history quite well: without the deeply rooted concept of infinity, the "sequence" part of the concept was the crucial part of the infinite sums. $\endgroup$ – orion May 20 '16 at 17:00
  • $\begingroup$ Isn't the German word Reihe? I wouldn't say that was equivalent to "order" (that would be Ordnung). $\endgroup$ – Henning Makholm May 20 '16 at 18:32
  • $\begingroup$ @HenningMakholm, in Russian the word for a series in math is ряд, while the word for order is порядок (in both biological classification and in ring theory), so it has ряд as part of it. I had not noticed that before. $\endgroup$ – KCd May 20 '16 at 21:52

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