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)

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)