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Is there a calculus (not analysis, not second-course in calculus) textbook that begins with sequences and series before moving on to functions of a real variable? I'm looking for a good dose of rigor too -- for instance, epsilon-delta definitions make an appearance but don't dominate the exposition, as is the style in most "honors calculus" books -- but probably any book that starts with sequences will have this.

It seems to me the main reasons calculus books start with limits of functions and then derivatives are inertia and curriculum constraints: Calc I is the terminal math course taken by some majors -- maybe biology, maybe business, etc. -- and that's they content those majors want.

I'm teaching a rigorous calculus course for math majors (but designed for students coming straight out of US high school, many with no prior calculus experience -- the average problem difficulty in Spivak seems a little too high, although the exposition is beautiful). I have the students for a whole year, so I'd really like to start them with sequences and series and move to functions of a real variable later. This is the more natural sequencing. But more importantly, students who have some prior calculus exposure will hopefully stay more engaged, or if not, at least realize immediately that they will need to get to work and cannot coast along on previous knowledge.

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  • $\begingroup$ So you want calculus + sequences and series + rigour, but you don't want analysis? What's the difference? $\endgroup$ – Will R Mar 13 '17 at 18:52
  • $\begingroup$ Honors calculus books, in my experience, don't go near multiple ways to describe completeness of the reals, uniform convergence of functions, point-set topology, and a quite a few other things that are, in some form, in pretty much every analysis book I've ever seen. The gap is quite large. Honors calculus books will still prove some things, introduce epsilon/delta without having them on 80% of the pages of the book, and might, for instance, prove standard facts about exponentials and logarithms, maybe even from the integral definition of $ln x$, include Lagrange's remainder theorem, etc. $\endgroup$ – Barry Smith Mar 13 '17 at 21:42
  • $\begingroup$ And the problems make a big difference. I am convinced that the median hours of experience with mathematical reasoning for college freshmen in the US is 0. Honors calculus books will not expect a formal proofs course. They'll have some problems that ask students to reason, but about formulas and graphs, maybe deriving identities or inequalities from others, or with open-ended or true/false questions about quantification, and in a few problems, truly abstract proofs, but short, using the simplest types of logical argument and intended to help students develop organic facility with logic. $\endgroup$ – Barry Smith Mar 13 '17 at 21:48
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Can I suggest Zorich's 'Mathematical Analysis'? It has analysis in the title, but I don't think it's anywhere near as bad as other books in it's vein. It contains all the standard material for a first course in calculus, and all the usual applications. It has a sequel which goes in the direction of vector calculus and other less-standard applications.

It's a little terse, but the exposition is rigorous and the problems are at a somewhat higher level than most you would see in first courses.

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You can check out this textbook.

http://www.springer.com/us/book/9780387305301

http://www.math.iitb.ac.in/~srg/acicara/

It starts with basics, but covers the content beautifully. It also has a sequel.

Hope this helps.

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You could try the old British classic by Burkill, A First Course in Mathematical Analysis (link is to legal free online version). From the sounds of it, they might need a bit of hand-holding due to the slightly old-fashioned writing style (and make sure they know a few Greek letters), but it's tried-and-tested and a well-respected book.

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