Meeting Details

Title: Grouping and rearranging terms in infinite series PMASS Colloquium Joe Roberts, Penn State When studying infinite series of real numbers, it is tempting to expect the familiar properties of addition to hold, and in some cases they do. Series that converge absolutely are as well behaved as finite series --- they are associative and commutative. However, series that are not absolutely convergent have very different properties. The Riemann Rearrangement Theorem states that any conditionally convergent series can be reordered to sum to any real number or to diverge (illustrating a failure of commutativity), and there are divergent series for which one can choose a subsequence of partial sums that converge to any arbitrary real number (a failure of associativity). I will give examples of these kinds of unexpected outcomes, eventually building to a construction due to Sierpinski of a single power series whose terms can be grouped to converge uniformly to any continuous function on [0,1] vanishing at 0 --- a "universal Taylor series".

Room Reservation Information

Room Number: MB113 04 / 03 / 2014 02:30pm - 03:20pm