Talk:Sum of Geometric Sequence

I'm partial to using sum notation, since unnecessary indefinite expansions with $\cdots$ seems somehow less exact since it's easier to lose track of something. I'm happy to defer to what others think, mostly I'm curious what the accepted custom is. I wrote out the proof with sums now while I was thinking of it.

Let $S_n = \sum_{j = 0}^{n - 1} x^j$.

Then $x S_n = x \sum_{j = 0}^{n - 1} x^j = \sum_{j = 0}^{n - 1} x\cdot x^j = \sum_{j = 1}^n x^j$.

Then $S_n(x - 1) = x S_n - S_n = \sum_{j = 1}^{n} x^j -\sum_{j = 0}^{n - 1} x^j = x^n - x^0 = x^n - 1$.

Thus $\sum_{j = 0}^{n - 1} x^j = S_n = \frac{x^n - 1}{x - 1}$ for $x \neq 1$.

--Cynic 03:32, 25 November 2008 (UTC)

Certainly, feel free to add it. I was in two minds about this one, the $\cdots$ style is elementary and simple to follow and may be accessed by students at an elementary level and is arguably more accessible. But no problem having both versions in. Thoughts? --Matt Westwood 06:23, 25 November 2008 (UTC)

... BTW I might make it go:

$S_n(x - 1) = x S_n - S_n = \sum_{j = 1}^{n} x^j -\sum_{j = 0}^{n - 1} x^j = x^n + \sum_{j = 1}^{n-1} x^j - \left({x^0 + \sum_{j = 1}^{n - 1} x^j}\right) = x^n - x^0 = x^n - 1$

to make it blindingly obvious ... --Matt Westwood 06:27, 25 November 2008 (UTC)

Why only standard number fields? Looks much more general than that. --Arthur 13:41, 25 June 2011 (CDT)


 * Because that was the context when I first put it together.


 * It's quite reasonable to address all these "arithmetic identity" sort of results in terms of the most general algebraic object that they can sustain, but it's always worth while keeping the readers' feet on the ground, so to speak, by providing them with the context in which it will be most immediately useful.


 * No matter, we can go through them all and apply them to the general field, if that's what is appropriate, but then you have to ask: does it apply only to complete spaces? We're using a limit approach here, there's infinite sequences going on, etc. etc. and then you find yourself going down some particularly abstruse metric-spatial and topological rabbit-holes that you start to think: perhaps this should be entered on a separate page - or else we should wait till be have built up the infrastructure to sustain the results.


 * So in short, yeah fine, we can do this, but perhaps we wait till we're clever enough. As for me, I ain't. --prime mover 14:25, 25 June 2011 (CDT)


 * Ok I disagree but won't bother adding such notes when it happens to involve convergent series. But this particular page is a purely algebraic result which obviously applies to any field, and in fact whenever 1-r is invertible in a commutative unital ring (not even ordered).


 * I am not familiar enough with analysis to know what requires any cleverness but my disagreement is because I am pretty sure you could get a significant improvement simply by identifying what subset of axioms common to "standard number fields" you are actually using and then stating the proof for the name that applies to that subset (eg topological ring or topological field) and simply mentioning each time that this includes any "standard number field". That would avoid repetition while still keeping the readers feet on the ground by providing the most useful context. At the same time it would help the reader grasp the appropriate levels of abstraction.


 * Abstruse rabbit holes requiring cleverness should only apply when you are not able to use the SAME proof that you are using in a standard number field, ie when you cannot identify a subset of axioms that has a well known name, which you are using for the simple proof..


 * In this particular case the general result on geometric series is of wide interest. It could be useful to link the background info at [wikipedia]. Although not mentioned on that general interest page I think it would be appropriate in ProofWiki to state the convergent result for unital [Banach Algebras]. You already have a page for Banach space. Wikipedia explains a good motivation for doing this level of generality for quite a few elementary results (while explicitly mentioning that they consequently apply to "standard number fields"):


 * "Several elementary functions which are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric series remains valid in general unital Banach algebras. The binomial theorem also holds for two commuting elements of a Banach algebra."--Arthur 02:32, 26 June 2011 (CDT)


 * First things first we do not, in general, want to link to Wikipedia if we can help it (for various reasons: a) wikipedia is not infallible, b) wikipedia may change to move a crcial piece of info linked to, c) we don't just want to be a site of wikipedia redirects, etc. etc.)


 * I agree with your sentiments, though. What we could do (and what would make a repeatable template for other pages) is add a section which a) defines the precise (or as precise as we currently understand) conditions under which the result is valid, b) points out that the "standard number fields" or whatever the "usual" conditions are for this result to hold, and c) why relaxing any of those conditions causes that result not to hold (e.g. "it doesn't hold for a ring because you need an inverse" or something, yes I know that isn't necessarily an appropriate point to make in this case, that was just a general example).


 * That's a direction which some of our contributors have started taking: note the recent work on GCD, where we have made a start. But clearly this is a task which will be a long time ongoing: hunting down each of these standard simple algebraic identities and rattling the frame to see how hard we can shake it before it falls apart. And what it needs is someone who knows about (or is in the immediate position of being able to learn about) the abstract algebraic concepts which are needed. --prime mover 03:46, 26 June 2011 (CDT)