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

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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

Resolved

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 f_n(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 $t\in \mathbb {R}$

$\textstyle |\sin(t)|\leq |t|;\;\mathrm {e} ^{t}\geq 1+t;\;2t\leq 1+t^{2}$ and obtain

$\textstyle |f_{n}(x)|\leq {\frac {1}{2n}}$ for all $x\in \mathbb {R}$.

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)