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)

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)