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I am just wondering what good books there are on Sequences/Series, having to do with questions such as these:

Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

I'm currently checking out Counterexamples in Analysis, maybe that is fitting. Also reading Intermediate Algebra, by Richard Rusczyk and Mathew Crawford. There are some topics in there that focus on sequences and series.

Thanks.

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user164403
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There are lots of questions that are like the one you linked in different ways, but it seems to me that you are most interested in what I've heard called "additive combinatorics". The subject actually gets pretty technical pretty fast, but fortunately, some great expositors have been working on it. In particular there's a book by Tao and Vu.

http://www.amazon.com/Additive-Combinatorics-Cambridge-Advanced-Mathematics/dp/0521136563

However, although I haven't read any of it, I suspect it's rather advanced. I hope someone can answer with more elementary introductions.

I remember attending lectures on at the PCMI Summer Session in 2002 about colouring integers and avoiding "monochromatic" solutions to linear systems of equations which were very very good and elementary, and I wish I could find the lecturer in hopes the notes were written up somewhere, but cannot.

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  • Ah, I see. I thought the subject I was trying to cover was more so real analysis which I thought to be just too beyond my high school level. Thanks, I'll check out the book. – user164403 Sep 29 '14 at 11:22