[QUOTE=CRGreathouse;463539]Sequence is 1,2,3,...; series is 1+2+3+....[/QUOTE]

* I wonder, how true that is languages other than English. I am pretty sure the series equivalent in Persian which is pronounced as "sery" and is probably introduced from French, does not have the summation concept included, even in Mathematical context.

* I wonder, How recent is the summation notion associated with the word "Series" in English. It is safe to presume that the word "Series: predates the concept of summation of Series/sequences. I would put my money on only decades old at most.

[QUOTE=a1call;463548]* I wonder, how true that is languages other than English. I am pretty sure the series equivalent in Persian which is pronounced as "sery" and is probably introduced from French, does not have the summation concept included, even in Mathematical context.

* I wonder, How recent is the summation notion associated with the word "Series" in English. It is safe to presume that the word "Series: predates the concept of summation of Series/sequences. I would put my money on only decades old at most.[/QUOTE]

You know what they say about a fool, an their money ... From the wikipedia on series ( mathematics), "This paradox was resolved using the concept of a limit during the 19th century." aka the 1800's had infinite series paradoxes solved. The word series, meaning row, or chain, and spelled the modern way was from the 17th century, look at the etymology on Google. The development of calculus, in it's modern form, would not be possible without them.

[QUOTE="http://jeff560.tripod.com/s.html"]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."

John Collins (1624-1683) wrote to James Gregory on Feb. 2, 1668/1669, "...the Lord Brouncker asserts he can turne the square roote into an infinite Series" (DSB, article: "Newton").

James Gregory wrote to John Collins on Feb. 16, 1671 [apparently O. S.]: "I do not question that all equations may be formed by tables, but I doubt exceedingly if all equations can be solved by the help only of the tables of logarithms and sines without serieses."

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.[/QUOTE]

[QUOTE=a1call;463486]That is the case with all series operations including sum and product. That's no reason for such a concept not to exist.[/QUOTE]
The distinction is that taking powers is not associative.

[QUOTE=Nick;463583]The distinction is that taking powers is not associative.[/QUOTE]

It is indeed a distinction, but not a disqualifier. Same way it does not disqualify exponentiation as a valid function.

[url]https://math.stackexchange.com/questions/2351850/power-towers-and-notation-for-iterated-exponentiation[/url] might be of interest to a1call

[QUOTE]Sorry science_man_88 is a moderator/admin and you are not allowed to ignore him or her.[/QUOTE]

What a shame.

[QUOTE=a1call;463594]What a shame.[/QUOTE]

I would have said sham I might as well be brown as it could signal full of :poop: the blog area was mostly made as a trash bin at last check.

The real source of sm88's quotes is Jeff Miller's [url=http://jeff560.tripod.com/mathword.html]Earliest Known Uses of Some of the Words of Mathematics[/url], section [url=http://jeff560.tripod.com/s.html]S[/url], which indeed dates it to the 17th century. By 1700 the terminology was nearly universal, so it caught on quickly -- just a generation or so.

[QUOTE=a1call;463484]It's odd no one seems to have contemplated the concept of power-of-series before.[/QUOTE]

[QUOTE=a1call;463486]That is the case with all series operations including sum and product. That's no reason for such a concept not to exist.[/QUOTE]

You might find Knoebel's [url=https://www.jstor.org/stable/2320546]Exponentials Reinterated[/url] (which won two prizes for expository writing) interesting. I don't think it covers your exact case but you can probably use its methods to prove results about it.

[QUOTE=CRGreathouse;463612]The real source of sm88's quotes is Jeff Miller's [url=http://jeff560.tripod.com/mathword.html]Earliest Known Uses of Some of the Words of Mathematics[/url], section [url=http://jeff560.tripod.com/s.html]S[/url], which indeed dates it to the 17th century. By 1700 the terminology was nearly universal, so it caught on quickly -- just a generation or so.[/QUOTE]
Firstly, that is an amazing source. Secondly you are an amazing researcher.
Thank you.

However, I could not decipher the origin of association of summation in those references with the word series or its variations.
Would be willing to eat my hat (if I was wearing one:smile:) if otherwise shown.

[QUOTE=CRGreathouse;463612]The real source of sm88's quotes is Jeff Miller's [url=http://jeff560.tripod.com/mathword.html]Earliest Known Uses of Some of the Words of Mathematics[/url], section [url=http://jeff560.tripod.com/s.html]S[/url], which indeed dates it to the 17th century. By 1700 the terminology was nearly universal, so it caught on quickly -- just a generation or so.[/QUOTE]

on that same site there's :


[URL="http://jeff560.tripod.com/mathsym.html"]Earliest Uses of Various Mathematical Symbols[/URL]

and

[URL="http://jeff560.tripod.com/ambiguities.html"]Ambiguously Defined Mathematical Terms
at the High School Level[/URL]

as well as :

[URL="http://jeff560.tripod.com/stamps.html"]Images of Mathematicians on Postage Stamps[/URL]

seems an interesting set of links.