User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)


 * $u \ v \ \mathsf{u} \ \mathsf{v} \ \nu \ \upsilon$

Anyone else have a hard time distinguishing between $u$ and $v$? I would like it to look more like this, does it confuse anyone else? It seems PW doesn't have the upgreek package. --GFauxPas 07:49, 27 January 2012 (EST)


 * Nope. Multiple years of extensive TeX writing and reading have trained my eye. I agree that referenced $v$ looks more distinguished, but imagine it is hard to implement. --Lord_Farin 08:08, 27 January 2012 (EST)

Convergence
We have that $\displaystyle \lim_{n \to +\infty}a_n = 0$, by hypothesis.

To show that $\displaystyle \sum_{k=1}^n{n \choose k}\frac {{a_n}^{k-1}} {n^k}$ converges, observe that, for $n$ large enough:

am I on the right track? --GFauxPas 14:10, 5 March 2012 (EST)


 * Yes, that's about what I had in mind. Notice, however, that relabelling of indices applies only to the summation index and its boundary values, so it should be $a_n^k$ instead of $a_{n-1}^k$. If you have trouble doing it in your head, consider making a note somewhere using the adapted summation index $k'=k+1$, writing everything in terms of $k'$ and then lastly rename $k'$ back to $k$ again. --Lord_Farin 15:15, 5 March 2012 (EST)
 * But I need to get it into the form $\displaystyle \sum^{n-1}$, how do I do that? Or is there another way to use Sum of Geometric Progression? --GFauxPas 17:19, 5 March 2012 (EST)


 * Well, the upper index in particular indicates the largest value your summation index takes. Relabelling will in general indeed adapt the upper bound; the upper bound hence is to be thought about as a property of the summation index. I suggest you practice relabelling some more using pencil and paper; the process can be a bit elusive at first. --Lord_Farin 17:31, 5 March 2012 (EST)


 * Is there something else "relabelling" is called other than that? I'm trying to google it and I'm not getting much related to what we're talking about. In order to practice I have to have some idea of what I'm doing, otherwise I don't know what there is to practrice. Eh, I think I'll leave this to another time, when I can learn this from a source, I have enough to do right now with other things. Thanks though! --GFauxPas 17:42, 5 March 2012 (EST)


 * It might also be called 'changing the summation index' but that's longer ;). Two references I found are and . Hope those shed some light, and as always, I'm glad to help out. --Lord_Farin 17:52, 5 March 2012 (EST)