Riemann’s Rearrangement Theorem

-

 In 1827, Peter Lejeune-Dirichlet discovered this surprising result while working on conditions that ensured the convergence of the Fourier series. He was the first to notice that it is possible to rearrange the terms of certain series (now known as conditionally convergent series) so that the sum would change. Why is this result possible? 

*ma049.qxd (washington.edu)

Comments

Post a Comment

Popular posts from this blog

Compactness in Metric Spaces

-
Image
  The idea of compactness comes from the fact that these are the nice set next to the finite sets. i.e. It shares many properties of finite sets in a topological space. The first thing common between finite and compact sets is, these are closed and bounded. Every sequence has a limit point. A continuous image of a compact set is compact and many more. Compactness has a very close relation to completeness . Let X be a topological space then the definition of compactness is defined in the following way-: X is said to be compact if every open covering of X has a finite sub-covering. Heine-Borel Theorem -: Heine-Borel make our work easy for Euclidean spaces by generalizing that, ‘ K is a compact subset of R if and only if it is closed and bounded. ’ But above theorem doesn’t work well for all metric spaces , for example -: Take N with discrete metric. What is missing here? Completeness? No! Boundedness? No! Closedness? No! Then what? The thing that is missing is totally bounded
Read more

Generating set for various groups including S_{n}, A_{n}, GL_{n} (\mathbb{R}), SL_{n} (\mathbb{R}) etc.

-
As we know that all cyclic groups are generated by a single element but what about non-cyclic groups?    How far any non-cyclic group is in the number of generators to cyclic groups? After going through the article (link attached below) you will know some interesting facts as well as the answer to the above questions.  Generating Set for Various Groups Click on the above link.
Read more