# 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.

• I think Moore-Smith convergence might help. Feb 23, 2013 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. Feb 23, 2013 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. Feb 23, 2013 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. Feb 23, 2013 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. Feb 24, 2013 at 9:03

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?