In pure mathematics, a sequence is a list of terms, for instance $1, \frac12, \frac14, \dots, \frac{1}{2^k},\dots$, and a series is the sum of an infinite sequence, for instance $\sum_{k=1}^\infty \frac{1}{2^k}$.

However, in many applied contexts, a sequence of data is called a series, for instance time series in statistics.

How did this difference in usage originate?

  • 4
    $\begingroup$ My guess is that the terminology distinction between "sequence" and "series" that is so prevalent in mathematics, and which is especially emphasized in mathematics textbooks since the 1960s or so (my guess), did not sufficiently overshadow the natural language usage that conflates these two terms when the term "time series" began being used. Note that there is a distinction between talking about the number $4$ (equals $5-1,$ IV, $\sqrt{16},$ etc.) and the numeral $4$ and the digit $4,$ but even in ordinary mathematical exposition these terms are often conflated. $\endgroup$ – Dave L Renfro May 19 '20 at 7:45
  • $\begingroup$ @DaveLRenfro Interesting -- so you suggest I should rather ask when mathematicians started to distinguish those two objects? $\endgroup$ – Federico Poloni May 19 '20 at 7:48
  • $\begingroup$ Couldn't a "series" also be the product of an infinite sequence (or finite sequence)? $\endgroup$ – Carl Witthoft May 19 '20 at 11:34
  • $\begingroup$ @CarlWitthoft I have never used with in that meaning. $\endgroup$ – Federico Poloni May 19 '20 at 11:51
  • 1
    $\begingroup$ Actually, a series is not defined as a sum. Otherwise, the expression "sum of a series" would be redundant, and "partial sums of a series" could not be defined at all. One needs to know the assignment of values to individual terms, i.e. the sequence, to define them. So, in the end, there is no difference between series and sequence as defined, except that the former term is more often used in the context of talking about summation. Considering the ubiquity of moving averages, etc., in statistics the conflation of sequence/series there is not surprising. $\endgroup$ – Conifold May 19 '20 at 17:16

As @Dave L Renfro noticed, the distinction between series and sequence is not old, and it was possible for the same author to use the two terms with different meanings (also in the same article). Consider e.g. Gauss's Theoria Residuorum Biquadraticorum Commentatio Prima & Commentatio Secunda, we have some examples:

  • article 5

cunctos numeros $1, 2, 3 \ldots p-1$ inter quatuor series A , B , C , D ita distribui, vt ...

all the numbers $1, 2, 3 \ldots p-1$ divide themselves into the four series A, B, C, D, such that ...

here A, B, C and D are (obviously finite) subsets of $[1,p-1]$

  • article 8

Sit $g$ radix primitiva pro modulo $p$, i. e. numerus talis, vt in serie potestatum $g$ , $gg$ , $g^3$ ... nulla ante hanc $g^{p-1}$ vnitati secundum modulum $p$ congrua evadat.

Let g be a primitive root for the modulus $p$, i.e. a number of such a type that in the series of powers $g$ , $gg$ , $g^3$ ... no power before $g^{p-1}$ will be congruent to unity relative to modulus $p$.

here series means the (finite) sequence of the power of $g$ from $g$ to $g^{p-1}$

  • article 37

Quum singulae normae sint vel numeri primi reales (e serie 2, 5, 13, 17 etc.)

Since the individual norms are either real prime numbers (from the series 2, 5, 13, 17, ...)

here series is the infinite sequence of prime numbers with a prescribed form

  • article 73

Disponamus partes ipsius $\varphi(2b,a-b)$ sequenti modo \begin{equation*} \begin{split} &[\frac{a-b}{2b}]+[\frac{3(a-b)}{2b}]+[\frac{5(a-b)}{2b}]+ecc+[\frac{(b-1)(a-b)}{2b}]\\ +&[\frac{a-b}{b}]+[\frac{2(a-b)}{b}]+[\frac{3(a-b)}{b}]+ecc+[\frac{\frac{1}{2}b(a-b)}{b}]. \end{split} \end{equation*} Series secunda manifesto fit $=\varphi(b,a-b)=\varphi(b,a)-1-2-3-ecc-\frac{1}{2}b=\varphi(b,a)-\frac{1}{8}(bb+2b)$ ...

If we order the terms $\varphi(2b,a-b)$ in the following way: \begin{equation*} \begin{split} &[\frac{a-b}{2b}]+[\frac{3(a-b)}{2b}]+[\frac{5(a-b)}{2b}]+ecc+[\frac{(b-1)(a-b)}{2b}]\\ +&[\frac{a-b}{b}]+[\frac{2(a-b)}{b}]+[\frac{3(a-b)}{b}]+ecc+[\frac{\frac{1}{2}b(a-b)}{b}]. \end{split} \end{equation*} the second series will evidently be equal to $=\varphi(b,a-b)=\varphi(b,a)-1-2-3-ecc-\frac{1}{2}b=\varphi(b,a)-\frac{1}{8}(bb+2b)$ ...

here series is a finite sum (whose result Gauss calls "summatio")

In Summatio quarundam serierum singularium:

  • article 6

Duas iam progressiones considerabimus [...] Progressio prima haec est $$ 1-\frac{1-x^{m}}{1-x}+\frac{(1-x^{m})(1-x^{m-1})}{(1-x)(1-xx)}- \frac{(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}+ecc $$ sive $$ 1-(m,1)+(m,2)-(m,3)+(m,4)-ecc $$ [...] obvium est quoties, $m$ fit numerus integer positivus hanc seriem post terminum suum $m+1^{\rm tum}$ abrumpi [...] adeoque [...] summam esse debere functionem finitam integram ipsius $x$

Now we will consider two "progressions" [...] The first one is $$ 1-\frac{1-x^{m}}{1-x}+\frac{(1-x^{m})(1-x^{m-1})}{(1-x)(1-xx)}- \frac{(1-x^{m})(1-x^{m-1})(1-x^{m-2})}{(1-x)(1-xx)(1-x^{3})}+ecc $$ or $$ 1-(m,1)+(m,2)-(m,3)+(m,4)-ecc $$ [...] it is clear that if $m$ is a positive integer, then this series end after the $m+1^{\rm th}$ term [...] and so [...] the sum must be a polynomial in $x$

here series (or progression) is a general sum, finite of infinite

  • article 8

sed propter rei singularitatem etiam de casibus iis ubi $m$ vel fractus vel negativus est pauca adiecisse haud poenitebit. Manifesto tunc series nostra haud amplius abrumpetur, sed in infinitum excurret, facileque insuper perspicitur, divergentem eam fieri, quoties ipsi $x$ valor minor quam 1 tribuatur, quapropter ipsius summatio ad valores ipsius $x$ qui sin maiores quam 1 restrinci debebit

However, for the particularity of this argument will not be unpleasant if we add a few words also about the case in which $m$ is fractional or negative. In these cases, evidently, our series is not interrupted, but extends to infinity, and moreover it is easily seen that it is divergent every time we assign to $x$ a value less than 1, which is why we have to limit the sum to the values of $x$ that are greater than 1.

finally here Gauss uses the term series in its "modern" meaning of infinite sum (either convergent or divergent).

Also consider that (in latin, of course) "time series" means a sequence of items ordered in time, so it is used in statistics with its original meaning:

Virgilium vidi tantum: nec avara Tibullo / tempus amicitiae fata dedere meae. / successor fuit hic tibi, Galle, Propertius illi; / quartus ab his serie temporis ipse fui. (Ovid, Tristia, 4, X, 51-54)

Vergil I only saw, and to Tibullus greedy fate gave no time for friendship with me. Tibullus was thy successor, Gallus, and Propertius his; after them came I, fourth in order of time.

  • $\begingroup$ Not to mention the even earlier usage "Omnia Gallia in partes tres divisa est" :-) $\endgroup$ – Carl Witthoft May 19 '20 at 11:53
  • $\begingroup$ @CarlWitthoft I know ... :-) but there is no reliable English translations of such papers. $\endgroup$ – user6530 May 19 '20 at 13:15

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