Wikipedia:Reference desk/Archives/Mathematics/2009 September 3

= September 3 =

Econometrics
Does anyone have any suggestions for a good undergraduate level econometrics text book?--98.240.70.102 (talk) 00:02, 3 September 2009 (UTC)


 * I used Gujarati's Basic Econometrics, a very user-friendly introductory book, if somewhat lacking up-to-date treatment of recent topics such as the asymptotic (large-sample) approach. A more modern choice can be Woolridge's Introductory Econometrics. Pallida  Mors  18:28, 3 September 2009 (UTC)

Binomial Expansions
I have to determine the expansion in powers of x up to $$x^4$$ of $$(1-x^3)^6(1-x)^{-6}$$. Now I can do this by some simple division and then expanding $$(1+x+x^2)^6$$ but is there any way of reaching the same answer by expanding the two sets of brackets separately? Thanks 92.4.122.142 (talk) 12:47, 3 September 2009 (UTC)
 * Yes, expand the first factor as $$1 - 6x^3 + ...$$ and the second as $$1 + 6x + (-6)(-7)x^2/2 + ...$$ Ignore powers of x above 4, and then multiply out the resulting expressions. Tedious, but it gives the same answer as your "simple division" method. Caution, expressions may contain typos. AndrewWTaylor (talk) 13:55, 3 September 2009 (UTC)
 * Exactly! See Newton's generalised binomial theorem. (r = – 6 is the case of the second bracket)  Dr Dec  ( Talk )    17:56, 3 September 2009 (UTC)

Uniform convergence of exp(-x2)sin(x/n) over R
Hi there guys,

could anyone please suggest a test or approach to check whether convergence of $$f_n(x) \, = \, e^{-x^2} \, \sin(\frac{x}{n})$$ to $$f(x)\, = \, 0$$ over $$\mathbb{R}$$ is uniform? I've tried everything I could think of (not much sadly) such as checking that obviously both $$f_n(x)$$ and $$f(x)$$ are continuous, and trying to find a value for the maximum of $$e^{-x^2} \, \sin(\frac{x}{n})$$, from which all I got was an ugly formula in xtan(x/n) (for x, not for fn(x)), and I don't seem to be making any headway. I don't need to be walked through what to do, but if I could just get myself aimed in the right direction that'd be great (e.g. the name of a test or a property to look at) - thanks!

Spamalert101 (talk) 18:20, 3 September 2009 (UTC)
 * No special machinery is needed here. To make f_n bounded by epsilon, just choose M large enough that exp(-x2) is less than epsilon outside [-M,M] and then make n large enough that sin(x/n) is smaller than epsilon inside [-M,M]. Algebraist 18:28, 3 September 2009 (UTC)


 * Also, you may use these elementary inequalities for all $$t\in\R$$
 * $$\textstyle|\sin(t)|\leq|t|;\;\mathrm{e}^t\geq 1+t;\; 2t\leq 1+t^2$$ and obtain
 * $$\textstyle|f_n(x)|\leq \frac{1}{2n}$$ for all $$x\in\R$$.
 * However, the first answer also shows a general principle (uniform convergence on intervals, with domination by a function vanishing at infinity, implies uniform convergence over R).
 * --pma (talk) 22:09, 3 September 2009 (UTC)

Thanks guys, that's a great help - didn't realize it could be that simple! Spamalert101 (talk) 14:02, 5 September 2009 (UTC)