As may be apparent from my latest blog entries, I have spent a decent part of my spare time lately on game theory. In addition to that, I have also been reviewing my university mathematics books, actually the one from our very first course, and at the time of writing this I am fascinated by sequences, series and limits.
During my university studies I did not really understand the beauty and essence of mathematics. Consequently, I did not put in the hours I should have, which I regret. The good news is that it’s never too late to start, so I took out my university math book, already last summer actually, and have occasionally studied the material from the beginning and concentrated on understanding the concepts and constructing proofs. I was pleasantly surprised by two things. First, I could actually understand the provided proofs and construct some of my own, and second, I was surprised how my attitude towards and perception of university level mathematics had changed: I could appreciate the thoroughness and logical flow of proofs and the joy of finding these properties of numbers and sequences. I have also understood better, how “inexact” methods, e.g. estimating a series from above, are permissible, often powerful methods that can provide exact answers.
In this and the following post I provide in total two proofs of my own, based on the practice problems from my old mathematics book. And yes, I must learn to use LaTex if I am to publish more of these posts with mathematical notation.