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)