Talk:Geometric series

Convergence - misleading?

"converges when |x| < 1, because in that case xk tends to zero" — the reader may conclude that convergence to 0 of terms of a series implies convergence of the series (that is, of partial sums), which is of course false (harmonic series is the simplest counterexample). Boris Tsirelson 22:20, 9 January 2010 (UTC)

Thanks for the pointer. I have just started to edit this page and intend to make a few changes. You are right that it may be misleading, however, it obviously was meant to explain why the limit of the sum tends to a/(1-x). --Peter Schmitt 23:16, 9 January 2010 (UTC)

"diverges definitely" seems to be a neologism; is it? Boris Tsirelson 12:50, 10 January 2010 (UTC)

Well, in German it is very traditional to say "bestimmt divergent". May be it was careless to translate this. Thank you, it is good to know that others check what one is writing. --Peter Schmitt 13:08, 10 January 2010 (UTC)

Series infinite?

Peter, I see that you completely rewrote this article, giving some explicit proofs. I also see that for you a series is necessarily infinite. I agree that in a more advanced context series are usually infinite, but in more elementary (high school) maths they can be finite. I have here the Collins dictionary and it states: series (maths) finite or infinite sum of terms. Abramowitz and Stegun define a (finite) arithmetic progression and write "the last term of the series is a +(n−1)d". In your definition the term "infinite series" would be a pleonasm, but I don't have to tell you that one meets the term frequently, I even own a book called "Infinite Series".

From WP :

A geometric series is the sum of the numbers in a geometric progression:

$\sum _{k=0}^{n}ar^{k}=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n}\,$

In Atlas zur Mathematik the name geometrische Reihe (consisting of n terms) is given, so in German, too, a Reihe can be finite. Hence, IMHO we should at least mention the elementary meaning of the term. One more thing: I have the impression that the term "ratio" is more common than "quotient" in the context of series. For instance, I believe that d'Alembert's convergence criterion is called the "ratio test". WP uses r and calls it ratio. --Paul Wormer 10:42, 10 January 2010 (UTC)

Paul, I hope you do not mind the rewriting. I thought that the article deserved some extension, and that led to changing most of the article. (I hope I found a good presentation.) Since the "proofs" are so elementary and short, I think that we should not resort to the "it can be shown" phrase. It even is not necessary to mention the binomial theorem.

As for "finite": I am aware of this, but I thought that it is used very rarely and is essentially old-fashioned. I may be wrong. Is the book on "Infinite series" the book by Knopp? The use of "infinite series", even if a pleonasm, may also be considered as either "tradition" or as stressing it because "series" alone is a little short.

As for school usage: There are also some "bad habits" in school that should not sustained (but clarified). If "finite series" is only used as synonym for "sum of a sequence" then this would be a bad habit. We have to say "the sequence of partial sums" of a geometric sequence. I'll think about how to do it -- but you may go ahead, of course.

Ratio: For me there is a slight difference between "ratio" and "quotient". I would use ratio mainly in the context of "proportion" (and "geometric progression") and "quotient" for a number (like the x or q here).

--Peter Schmitt 12:38, 10 January 2010 (UTC)

Another quote from WP, however: "In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·." [1] Boris Tsirelson 12:54, 10 January 2010 (UTC)