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Re: Sequence convergence and absorption
Posted:
Dec 2, 2009 12:02 PM


Robert Burn wrote: > One futher thought. When discussing real sequences, the notion of BALL > does not sound right. If you graph a sequence by plotting the points > (n, a_n) then convergence is captured by sleeves parallel to the > xaxis.
I strongly suggest plotting instead (1/n, a_n). For clarity, label the xaxis with the values of n.
This puts the visual weight on the tail of the sequence., which is what is important for convergence. The standard graph (n, a_n) puts equal visual weight on everything, hence nothing stands out visually. And unfortunately it emphasizes the beginning of the sequence  exactly the wrong part.
A major benefit of this alternative graph is that many students can see the lim sup and lim inf directly, although others will need to do some work to connect the picture to setbased definitions of the Tail Sup (pun intended) sequence TS_n = sup { a_m : m > n } and otherwise lock down the ideas.
I'm still considering the whole "absorbing set" idea. For example, intersect all the absorbing sets and now talk about lim sup and lim inf of a sequence. How do students handle that approach? I wonder what whacky  ummm  unorthodox interpretations they'll generate...
David Olson Michigan Tech



