Putnam Analysis
An extension of the Sums unit --- rather than just swapping infinite sums, we now get to enjoy swapping infinite integrals as well.
Contents and Brief Overview
This unit is a follow up to the sums unit, in which we shift the focus from swapping infinite sums to swapping infinite integrals. This is one of the more “hi-tech” OTIS units.
The unit starts off by dealing with absolute and conditional divergence and moves onto some “special” infinite sums, mainly Fubini and Tonelli, for double indexed infinite sums. The unit then discusses about the Riemann Integral. The next topic which is discussed is the so called method of “discretization”.
The next big thing in the unit is Lebesgue Integrals which, being a much more involved topic isn’t talked about in detail in the unit itself. Swapping double integrals and Converging of Taylor series show up next. The unit ends with an optional but rather interesting discussion on Fourier Analysis and the Stolz-Cesàro Theorem (I definitely didn’t copy paste the spelling).
Pre-requisites
As said earlier, this is a pretty technical unit. Pre-requisites include: Sums, Analysis (Preferably the Z version)
Notable Problems
- Putnam 2015 B6
- Putnam 2004 B5
- Putnam 2013 A6
- Putnam 2010 A6