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I want to learn everything about sequences, sums and products from A – Z. Is there one book that stands out from the rest quality wise? I want to start my research off right!

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I assume you’re asking about real and complex infinite sequences, series and products. I don’t know of any text which gives anything like a “complete” or even a “cutting-edge” treatment of this topic.

One reason is that the subject of infinite series was much more mathematically fashionable in the period from, say, 1800 to 1900 than it is now. During that time if you were a leading mathematician it is a good bet that you devoted some of your work to infinite series. The only leading 20th century mathematician I can think of who really deeply studied infinite series *per se* is Ramanujan, who is of course an exceptional case in many (perhaps most) ways. I think though that part of Ramanujan’s story is that an absolutely brilliant mathematician who devoted himself to the theory of infinite series already seemed a bit strange in 1912-1913, when Ramanujan sent letters to several Cambridge mathematicians. And of course Ramanujan had a tragically brief life and career. The last mathematician I can think of who devoted himself wholeheartedly to infinite series over the course of a long career is Alfred Pringsheim. It is interesting to read about Pringsheim (in some ways other than his mathematics, in fact) and his work. One gets the sense that although he was highly respected in his own time, from the vantage point of history he was mostly polishing the more fundamental discoveries of earlier mathematicians, and that he might have done better to devote himself to a different topic.

Nowadays the study of infinite series (and sequences and products) comes up naturally in analysis and plays an important part, but is not really a flourishing discipline in its own right, so far as I know. More pertinently, the best way (I think) to learn about infinite series is to continue one’s study of real and complex analysis. For instance in calculus one learns about power series, but to study them in any depth one wants to have a theory of convergence of sequences and series of functions, which is learned in undergraduate analysis (concentrating on uniform convergence) and refined in graduate analysis with the tools of measure theory and integration (allowing one to say nontrivial things about pointwise and $L^p$-convergence, among others). Similarly, the theory of complex power series is basically synonymous with that of functions of a complex variable, so one should take undergraduate and then graduate complex analysis rather than a course on complex series *per se*. If you want to learn more about Fourier series than their definition you should take measure theory and then a course in harmonic analysis, and so forth.

But okay, you asked for a book. I have owned Konrad Knopp’s Theory and Applications of Infinite Series since my undergraduate days and still find it to be reliable and enriching. It’s not intense though: as it concentrates on series and not the broader mathematical tools described above it can only go so far.

There is also a recent book on infinite series by D. Bonar and M. Khoury. This MAA text is very nicely written but is at a lower level than Knopp’s text. It is very far from being encyclopedic: even these brief, rather scattershot lecture notes of mine contain some material that the text by Bonar and Khoury omits.

Finally, let me say that I think this is still a good question and I wish that the state of affairs were not so extreme: the mathematical community as a whole seems to be in danger of forgetting much of what it once knew about infinite series. The treatment of infinite series (including series of functions) in Rudin’s *Principles of Mathematical Analysis* is beautiful — the high water mark of one of the great American math texts — but it seems to have been *too* influential: it is now very hard to find an analysis text which covers topics that Rudin’s text does not. This seems extreme. For instance, I never learned the Euler-Maclaurin Formula (really: I still don’t know it well!) because it never appeared in any of my courses or required texts. (It does appear in Knopp’s book though; I’m not sure about Bonar and Khoury.) That seems like a shame.

*Generatingfunctionology* by Herbert S. Wilf available for free here explains sequences and sums well.

*Difference Equations* by Walter G. Kelley and Allan C. Peterson is highly recommended for understanding math behind summations well.

*Theory and Application of Infinite Series* (Dover Books on Mathematics) Paperback

by Konrad Knopp

But it would be more helpful if you say what level you are starting from, and what sort of research you are contemplating.

3rd chapter of Rudin’s Principles of Mathematical Analysis for sequences. http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

I am not sure what you mean by sums and products, but if you are interested in Algebra, how about something like http://www.amazon.co.uk/Basic-Algebra-Groups-Rings-Fields/dp/1852335874 ?

If you want some thorough treatment , I would advise Algebra An Elementary Text-Book(PART II) by G Chrystal. This is perhaps the one book which treats the subject in its highest form from a general point of view. And please do not go by name , cause ‘elementary’ here just means that its not modern algebra.

Also , Infinite Series by Bromwich is excellent.

Knopp Infinite Series is equally good. But I think you should look through all the books and compare the proofs etc. If you reach a considerable level you might wanna try Divergent Series by GH Hardy ! Any book by Hardy is must read.

I’ve seen that the mentioned references are for undergraduate students, for research references see **Collected Papers volume V – Serge Lang with Jay Jorgenson**, they deal with theta and zeta series and delta products, seem’s to be a generalization of properties of the Riemann zeta function, some of them can be extended for series on eigenvalues of Frobenius acting on l-adic cohomology, eigenvalues of the p-form Laplacian or of pseudo-differential operators acting on Riemannian Manifolds.

I read Introduction To The Theory Of Infinite Series (1908) by T. J. Bromwich as a Dover Press re-print while in high school. It is still avalailable. I recall the title of my library copy was “…… Infinite Sequences And Series.” No calculus or higher algebra, so almost no pre-requisites required. A lot of good basic material, clear and concise.

Sample : (1).If $a_n\geq 0$ for all $n$ then $$\sum_{n=1}^{\infty}a_n<\infty \iff \prod_{n=1}^{\infty}(1+a_n)<\infty.$$ (2). If $1>a_n\geq 0$ for all $n$ then $$\sum_{n=1}^{\infty}a_n<\infty \iff \prod_{n=1}^{\infty}(1-a_n)>0.$$

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