Diskuto:Serio (matematiko)

Serio = vico de nombroj & konverĝa ≠ konverĝa

According to me, the content of lemma Serio matematiko comes close to my conclusion on 'the definition of series' : the problem is not the word series (always 'number sequence'), but the double meaning of 'convergent'.

Trying to find a proper description of how the word series is used in practice (not in 'defining' chapters in textbooks), my result is:

When the sum-sequence of an infinite succession, given by a law, of numbers, is studied (its clustering and limit), the succession is mostly denoted verbally by series (série, Reihe), and symbolically by a formula with the sigma sign or with plusses between the first few terms.

A worldwide convention says: if a succession is denoted by the word series, the adjective 'convergent' (or forms of the verb 'to converge') means 'clustering of sums' instead of 'clustering of terms'.

Isn't this enough for a math student for the rest of his/her life? Maybe with mentioning alternative descriptions, as there are:

+ The word series stands for an expression of a certain kind (in fact two, or more?).

Comment. Wheras nobody ever has explained how an expression can converge.

+ The phrase <series with general term x_n> (1) stands for: <sequence of partial sums of the sequence with general term x_n> (2) .

Comment. Despite the fact that the terms of (1) differ from the terms of (2) .

+ The word series stands for: the combination of a number sequence and its sequence of partial sums (Dieudonné/Bourbaki 1942) .

Comment. A curious way to say: by tradition, a succession of numbers is called series when (the clustering and the sum of) its sequence of partial sums is at stake.

Maybe mentioning from the history of 'convergent':

Long ago, the verbal denotation of a succession of numbers by 'convergent series' , could mean (e.g. by Gauß): clustering of the terms at limit zéro.

After Cauchy's Cours d'Analyse (1821), up untill now, the only meaning of 'convergent series' is: a succession of numbers with clustering partial sums.

In the decades around 1900 the idea came up Charles Meray 1872 – Konrad Knopp 1920) that the clustering of the terms of an infinite succession is more fundamental than the clustering of its partial sums. Instead of introducing a new word, the meaning of 'convergent' changed over to clustering terms, in situations where an infinite succession isn't denoted by series but by the quite new sequence (Folge, suite/variante).

Unfortunately 'convergent series' didn't change into 'summable series', although 'summable' allready was in use – at least in German – as synonym for 'convergent' for a long time (summirbar oder convergent).

Needless to say that it will be difficult to describe the meaning of series, when you're not aware of the fact that the meaning of 'convergent', depends on the choice for the name sequence or the name series .

To what extend there will be agreement with the above-mentioned?

Surprising The lemma Konverĝa serio starts with: "En matematiko, serio estas sumo da eroj el vico de nombroj". Consequently the number with the verbal name <one> and the symbolic notation < 1 > is - in Esperanto - a series. For this number is the sum of the sequence (2^-n) . Hesselp (diskuto) 20:32, 16 jan. 2023 (UTC)