This will be a talk for the Mathematics Colloquium at the University of Warwick, to be held October 19, 2018, 4:00 pm in Lecture Room B3.02 at the Mathematics Institute. I am given to understand that the talk will be followed by a wine and cheese reception.
Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.
This talk is based in part on joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.
- The rearrangement number. [bibtex key=BlassBrendleBrianHamkinsHardyLarson:TheRearrangementNumber]
- The subseries number. [bibtex key=BrendleBrianHamkins:The-subseries-number]
- Talk notes
How can a rearrangement change the value of a sum of real numbers, because addition of real numbers is symmetric [ a + b = b + a ] and associative [ (a + b) + c = a + (b + c) ]?
Rearrangement does not affect the sum of a finite sum, for the reason that you mention. However, it can affect infinite sums, in certain cases, and this is precisely the content of Riemann’s rearrangement theorem. It is not difficult to argue that rearrangement does not affect infinite sums consisting entirely of positive numbers, since the finite partial sums of the series and the rearranged series will be cofinal in each other and therefore have the same supremum. And similarly with all-negative sums, and with infinite sums that are absolutely convergent.
But in the case of sums that are conditionally convergent, however, then rearrangements can change the value of the sum. A conditionally convergent series is an infinite sum that converges, but not when you take the absolute values of the terms in the series. Thus, the positive and negative terms separately diverge to infinity (and negative infinity), but taken together, they cancel and the sum converges. In this case, Riemann proved, you can rearrange the terms to make the sum whatever you like. The way the proof goes is this: take a few positive terms until they add up to more than your target value, then take negative terms until you’ve undershot the target, and then positive terms again and so on. In the limit, you will hit the target exactly.