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I have a reasonable background in undergraduate real analysis, and I am very interested in questions concerning the convergence of series, and also how this relates to irrationality measure. I would like to know what the current state of modern research on convergence is, my end goal being to fully understand all of the research in this area. I have encountered a few papers online that contain very interesting results that establish the convergence of, or convergence tests for, a wider class of series (relaxations of the monotonicity requirement in the integral test, for example), and have seen a few things about how the convergence of series can relate to irrationality measure. Unfortunately, I have found that I generally do not have the prerequisites for reading these papers, and each one references so many others that I feel I don't have any grasp of what the overall research on the subject looks like. So, are there any good monographs or textbooks I can read to gain an understanding of modern research on the convergence of series?

To clarify, I already have an undergraduate-level education in real analysis, so I'm not looking for straight analysis textbooks, unless they have a heavy emphasis on convergence of series.

jeff honky
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    Regarding convergence and irrationality measure: is the Flint Hills series what you had in mind? – Ben Grossmann Feb 07 '20 at 20:25
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    The Theory and Application of Infinite Series might be a good place to start. – Clayton Feb 07 '20 at 20:26
  • @Omnomnomnom Yes; I have seen a paper that proves that its convergence implies that the irrationality measure of pi is less than or equal to 2.5, and I saw an MO answer here that mentions the irrationality measure of pi in the proof of the convergence of a series – jeff honky Feb 07 '20 at 20:32
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    Try the google search "thesis" + "irrationality measure" + "series". Other than books dealing more generally with transcendental numbers, I don't think you'll find any published books having a nontrivial focus on what you're asking about, but perhaps some Ph.D. or Masters theses might have this. There might also be some expository/survey papers, but you'd likely find those already cited in literature you know about, but probably not citations to theses, hence my suggested google search. – Dave L. Renfro Feb 07 '20 at 21:25
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    Incidentally, when I said "other than books dealing more generally with transcendental numbers", what I had in mind were books such as those I gave in my answer to Irrationality measure. – Dave L. Renfro Feb 07 '20 at 21:32

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