# The role of sequences in calculus

What is the role of sequences in (undergraduate, engineering) calculus? Is it just there for a more gentle introduction to limits, instead of starting with limits of functions (which is actually a core topic of calculus) and as introduction to (infinite) series?

In that case my other question is: Where do squences belong? (discrete math ?)

Or, are sequences a core topic of calculus?

note: I am aksing this question with disregard whether a topic is a prerequisite for a calculus course or is covered within. eg: I would not classify trigonometric identities nor hyperbolic functions as part of calculus, even though they are needed in a calculus course for integration techniques.

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I think Moore-Smith convergence might help. – Frank Science Feb 23 '13 at 12:58
@Frank: Trying to teach convergence of simple sequences in a typical U.S. first-year calculus class is hard enough; general nets are out of the question. – Brian M. Scott Feb 23 '13 at 13:00
@BrianM.Scott I have a calculus book, Курс дифференциального и интегрального исчисления, translated from Russian, introducing Moore-Smith convergence without prerequisite of any topology knowledge, even metric spaces. – Frank Science Feb 23 '13 at 13:16
@Frank: I don’t doubt it. It was written for a different kind of course, with different expectations concerning the students’ preparation and knowledge. It might be usable in a few honors calculus classes in the U.S., but in general it would be a disaster. – Brian M. Scott Feb 23 '13 at 13:27
I consider limits of sequences an essential topic in such courses. In my experience students do find them easier to grasp initially than the limit of a function at a point, so they serve a useful purpose early on, and they are certainly essential to understanding infinite series, which in turn are an essential part of any year-long calculus course, even one of a relatively cookbook nature. – Brian M. Scott Feb 24 '13 at 9:03

## 1 Answer

In the first place you should use sequences where they genuinely occur: In the definition of new objects, like $\exp$, as a tool to represent arbitrary (and maybe unknown) functions in a uniform way, as sequences of arrival times, as solutions of difference equations or more general recursions, and on, and on.

In my view sequences should be abolished as a means of understanding continuity or limits. Why would anyone test a gogol of sequences in order to prove a single instance of $\lim_{x\to\xi}=a$ when looking at the proper inequality makes the statement obvious. The problem with understanding limits is the handling of nested quantors. Why should you unnecessarily add to more of these when explaining what a limit is?

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There are no more quantifiers in the definition of the limit of a sequence than there are in the definition of the limit of a function at a point. The reason for introducing limits of sequences first is pædagogically quite straightforward: in general students seem to find it a simpler concept. Note that one need not (and in my experience does not) then go on to define the limit of a function at a point in terms of convergence of sequences, as your ‘test a gogol[sic] of sequences’ seems to suggest. – Brian M. Scott Feb 24 '13 at 8:59
@Brian M. Scott: Defining limits of functions via sequences adds at least two quantors. Papula has sold millions of calculus books for engineering students (in German) where limits as well as continuity of functions are defined via sequences. – Christian Blatter Feb 24 '13 at 10:02
I’m perfectly well aware of what’s required for that definition of continuity. But I said nothing about defining limits of functions via sequences. Indeed, I explicitly pointed out that this is not usually done in the approaches most commonly found in U.S. calculus courses of the kind under discussion. – Brian M. Scott Feb 24 '13 at 16:29