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)
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:
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.
- (Added 13:20, 10 January 2010 (UTC)) One usually only says: "The series converges" not the "infinite series converges".
- 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)
- (1) Indeed, a translation of Knopp (Dover). (2) We can mention that a "finite series" is better called a "sum of a finite sequence", but still signal the old-fashioned terminology. (3) I know that ratio is the same as proportion, but yet I have the impression that with regard to (geometric) sequences the term "ratio" is traditional. (4) I believe that our foremost purpose must be to describe what is in existence, what can be found in the (maths) literature. This is in contrast to a textbook author, who has the aim to educate young people, and can hope that a better nomenclature will give his/her readers a better understanding. --Paul Wormer 13:22, 10 January 2010 (UTC)
- I agree: We have to describe what is used. But I think we also should do it in a way that guides to the "better" usage (and explains why). Usually, the "higher" language makes subtle differences clearer, and pointing out this differences can help to better understanding. Many misconceptions are based on a vague (or completely wrong) interpretation of scientific terms. In mathematics (think of "imaginary", etc.) and certainly also in physics, etc.
- Yes, you were right: It is probably better to mention "finite" here, as well. (A full discussion is necessary in series article which seems to blur the issue, too. Can you agree with my recent changes? --Peter Schmitt 17:37, 10 January 2010 (UTC)
I avoided to include the "formula" of the geometric series since this is an elementary topic, and non- scientists often don't like formulas. The essential information can be given without it.
The reason for using q in the introduction and leaving the x later was that (as the section heading says) I wanted to stress the power series aspect (series of functions). The geometric series essentially is a single power series, not many series of numbers. Perhaps some sentence explaining this was missing, but I still think that it should be x.
Is the new last line in the examples useful? The partial sums can be calculated directly from the example. (Using the formula is more complicated.) Instead, perhaps the general terms for the series and the partial sums should be included.
--Peter Schmitt 11:06, 11 January 2010 (UTC)
- I added the formula to the lede; without an equation the lede is as good as void. I avoided the summation symbol. Somebody who cannot understand powers and plus signs won't read this article.
- I changed x to q because the last formula (the sum of the infinite series) contained a q and from q to x again to q seemed somewhat inconsistent. One could also use a x throughout (I am raised with r ).
- Yes, I believe that an application for the formula for Sn is useful, because bookkeepers use it for computing compound interests. I don't agree that direct summation of 5 fractions is easier. For 5 terms the ease is debatable, but for more terms the formula quickly wins. --Paul Wormer 11:53, 11 January 2010 (UTC)
- Of course, the formula wins. You don't need to tell me (I am a mathematician :-) I only meant that you do not the formula for adding the few numbers of the example.
- What computations of bookkeepers do you mean? For interest the geometric sequence is needed, not the series. --Peter Schmitt 00:54, 16 January 2010 (UTC)
- You're probably right, I never learned bookkeeping, but remembered seeing tables of Sn, must have been something else. --Paul Wormer 12:09, 16 January 2010 (UTC)
- PS After I wrote this I remembered that I still own a logarithm table from my schooldays. In the back there is a table with heading
- and it says something about capital and compound interest (in Dutch). So apparently there is some application. --Paul Wormer 12:15, 16 January 2010 (UTC)
- You are right, I also faintly remembered that I did computations using the geometric series (in "Zinseszinsrechnung"), but I could not remember for what. Now I found out: You need it when you want to calculate a (constant) payment rate on a loan from the intended time period. --Peter Schmitt 13:06, 16 January 2010 (UTC)