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Sequences and series are related concepts but differ extremely from one another. I feel that students in integral calculus frequently mix them up. Part of the problem is that:

  1. Sequences are usually taught only briefly before moving onto series.
  2. The definition of a series involves two related sequences (terms and partial sums).
  3. Both have operations that take in a sequence and output a number (the limit or the sum)
  4. Both have convergence tests for convergence (monotone convergence and squeeze theorem vs. root test, ratio test, etc.)

What methods can I use to teach students to distinguish between sequences and series? Specifically, methods that adress the above concerns.

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This is a problem I frequently have. If it is alright with the community, I'd like to start asking about specific concepts I've seen students struggle with. –  Brian Rushton Mar 27 at 23:42
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I think this is a great question. Asking about specific problems students have is exactly what this site can be for, I hope. –  brendansullivan07 Mar 27 at 23:46
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it is interesting that we teach Riemann sums before really giving a proper definition of sequence, much less series. Of course, there is geometric intuition, but, think about it. That is probably their first exposure to sequences in the standard American calculus sequence. –  James S. Cook Mar 28 at 2:04
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A good question. FWIW in these parts your first point does not apply. Here sequences are the first limit process the students encounter, i.e. first semester calculus. Series are second semester material. But the problem remains. Disturbingly many students still mix the two. –  Jyrki Lahtonen Mar 28 at 6:24
    
If they are in the US explain to them that every fall the top team from the American League and the top team from the National League meet for the World Sequence. They'll remember then. –  Jim Hefferon Mar 28 at 13:38

7 Answers 7

up vote 23 down vote accepted

I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term "series" phased out of usage in this context.

To many minds, "sequence" and "series" convey the same thing: a list of items. In modern parlance, we speak of a television series (a sequence of episodes) or a comic book series (a sequence of issues) or the World Series (a finite sequence of $n$ games where $4\leq n\leq 7$). Dictionary definitions for "series" cite everything but the mathematical definition, and even use all sorts of other mathematical terms like "group" or "number" or "set". No wonder students are confused! Only in math does "series" mean "sum of terms".

I am going to stop using this term, and instead refer to them as "infinite sums". It's exactly what they are, and it immediately triggers the fact that the terms are added, as opposed to being listed. I don't know whether this will solve the problem of students misunderstanding the differences between an infinite sum and its sequence of partial sums, but I know that this cannot possibly hurt their understanding.

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(+1) excellent answer. However, it is highly unlikely that phasing out 'series' will succeed. It is too widely used. I think that presenting series first by calling them 'infinite sums' is ideal, but one must also teach the students the common-place vocabulary. –  Ittay Weiss Mar 28 at 0:12
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Sure. I suppose when I said "in this context" I meant "when we first teach series to calculus students". Personally, I'll keep saying "infinite sums" forever, but I'll know what you mean if you say "series" :-) –  brendansullivan07 Mar 28 at 0:12
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I am neither upvoting nor downvoting this. I think you are absolutely correct in that a big part of the explanation is that natural language meanings of words occasionally lead students astray. But I disagree with your suggested remedy. Even if you succeeded in alleviating this problem, you would only postpone the problem to the next occasion where there is a clash between mathspeak and natural language. Undoubtedly you will be able to help a number of students with your practice, but I can't shake the feeling that it amounts to sweeping some dust under the rug. –  Jyrki Lahtonen Mar 28 at 6:39
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Indeed, it is fairly perverse to call infinite sums "series", and to formally declare colloquial synonyms to be radically different within mathematics. Although some people would argue that "infinite sums" do not exist, because we cannot add infinitely-many things, obviously this is the intention, no matter what technicalities come up. For more advanced students, yes, the more-complicated and chaotic reality must be exposed at some point, but portraying somewhat-ugly reality as an ideal is silly. –  paul garrett Mar 28 at 14:31
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@GitGud If you define "infinite sum" to mean the same thing as a series, then it has a meaning by itself. Every term was invented and defined by someone. –  Brian Rushton Mar 28 at 14:49

I second Andrew Sanfratello's answer—there's just no getting around the fact that technical terms have technical meanings which differ from their everyday meanings. And when I teach series, I start each class by writing "SERIES MEANS SUM" at the top of the board every day. (At one previous institution, I often taught in a room with several large boards, so I could easily keep that visible, in huge letters, for the whole lecture.) A silly gimmick, but it helped.

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LOL! Gotta try this. –  Jyrki Lahtonen Mar 28 at 6:47

Mathematics is littered with terms that are commonly used in the real world that can mean very different things. Sequence and series are just two. Group, function, kernel, field, and ring are just some examples of terms that have very exact meanings in various mathematical areas, but are used colloquially in English very differently.

If you can emphasize this idea, and the fact that sequence and series are two very exact definitions, I think you may find some more success.

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This isn't limited to mathematics, either. –  Mark Meckes Mar 28 at 6:12

Historically it seems that "sequence" is the interloper. "Series" was used to denote both concepts going as far back as Wallis, usually with the qualifier "infinite"; sometimes "progression" is used to denote what we call a sequence. Our current use of "sequence" in the theory of series did not become common until later, perhaps around 1900, as did the prominence of the term "partial sum." In modern terms, one might say that a sequence or a series is simply a function $f \colon {\bf N} \rightarrow {\bf R}$ -- the same in both cases. I think the significance of these observations is that a sequence and a series are not (readily) distinguished by what they are. They are distinguished by how they are used. We speak of the "sum" of series and the limit of the "terms." The words "sequence" and "partial sum" began to be used, I suppose, to help clarify the intended use. You do not sum a sequence, and you do not find the limit of a series; but you do find the limit of the sequence of partial sums of a series. To me, the notion of a sequence and a series are intrinsically difficult to keep straight, the capital sigma being the main difference (or the plus dot-dot-dot).

Difficult is not the same as impossible. Certainly "the sum of a series" and "the limit of a sequence" are easily distinguished. I warn my students ahead of time that synonyms are going to be used for different things, and that they will have to make an effort to learn the difference. I probably repeat it a few times, and I repeat it anytime someone makes a mistake in class. There did not seem to be a big problem with the terms. Unpacking the definition of the sum into the limit of the partial sum causes the most difficulty (leaving aside the other complexities of the topic). I do a few things that probably help the students learn the meanings: quizzing and testing on the definitions and testing them on the proofs of the n-th term test and integral test.

I'm not sure how much "convergence" adds to the potential for confusion. It certainly adds some, but perhaps only a little. The terms are used analogously for improper integrals, so the connection should make it easier to learn their meanings. One approach, which did not catch on, is to start first-year calculus with sequences (see Courant, rev. Courant and John, Intro. to Calc. and Anal., or Hardy, A Course of Pure Math., for examples). Then sequences are a long way from series in the development of the course. I suppose getting to derivatives quickly, for the sake of science and engineering students, is the reason for the current popular sequence of topics.

I do find that students' performances on the first test on sequences and series to be the lowest on average. There are many reasons. One is that for the students from AP Calc AB courses, it's the first test for which they have to learn new concepts. Another is that most of the problems have a different nature -- show whether this series has an answer rather than find the answer. Finally there is the range of tests of different natures: some are for divergence, some are for convergence, and some are for both. Nomenclature is definitely a factor but not the major factor. This probably depends on how the subject is taught, though. Perhaps I have minimized the impact of sequence/series confusion at the expense of something else.

I do think we owe it to students to teach them to be conversant with mathematical lingo. I tutored a middle school student, whose teacher did not use the word "cancel." The student in all seriousness suggested that the next step was to "s----" it. It sounded like a mix "smush" or "scrunch" -- I don't recall clearly now. It was a made-up word. That seems extreme to me. In any case, language evolves so perhaps the lingo will change. Perhaps it's worth pointing out that the root of the meaning of "series" comes from a word meaning to join up. So "series" means the terms "joined up" (by addition) to form a whole. Sequence, of course, comes from a word meaning to follow and implies that the terms are viewed as separate things. One might use the etymology to help students think about the meaning of the mathematical terms.

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I particularly like the statement "They are distinguished by how they are used.". I try to keep the focus on the question that we're trying to answer (sum or limit). –  Loop Space Mar 31 at 10:11
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+1 for "we owe it to students to teach them to be conversant with mathematical lingo". I might go even further and say that we owe it to students to teach them (through examples) how to deal with poor terminology that isn't going away. –  Mike Shulman Apr 8 at 16:06

It is unfortunate the term convergent has radically different meaning as it applies to sequences and series. A sequence $a_n \rightarrow L$ as $n \rightarrow \infty$ essentially means we can capture the tail of $a_n$ within as small an epsilon band as we desire. Whereas the series formed from $a_n$ converges iff the sequence of partial sums converges. Of course, we who are educated understand the convergence is in both cases the convergence of sequences. But, the fine print is often missed by the starting student who doesn't understand that $a_n$ converging to $1$ does not mean the series $a_n$ converges to $1$. Of course, I am being deliberately careless in my use of the term series in this post because this is how the abuse is accomplished in the minds of calculus students. If we used a wholly different term for convergence of series this would help separate the concepts.

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The term "summable" is fairly standard. I would always use that instead of "convergent" when referring to series. –  Neil Strickland Mar 28 at 8:13
    
One could also use a different term for convergence of sequences, e.g. talk about a sequence "having a limit" rather than "converging". This matches better how we use terminology for functions earlier in calculus: we talk about a function "having a limit" as $x$ approaches some number, but we talk about an improper integral of that function "converging". –  Mike Shulman Apr 8 at 16:03

I suggest that this is more than a problem in wording. In mathematics, many objects are introduced as a process (e.g. functions as "making a y out of a given x", sets as "taking things together") but later need to be used as an object (e.g. derivation of functions, sets of sets where the "inner" set must be perceived as an object (topology, measures)). This also seems the case here. Students maybe can see a sequence as a process, but not as an object. So the sequence of partial sums is too complex for their thinking as it includes infinitely many (finite) sequences. Quite often, "objects of objects" don't work with undergraduates: Operators in functional analysis (functions of functions), topologies (sets of sets), conditional statements in logic (statements of statements), ...

If you think this might fit your situation, have a look at this paper where APOS-theory which focusses on process-object duality is applied to series and sequences.

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+1 I really like this answer, too. I do feel like there's something more than just the nomenclature issue, and this seems to be it. –  brendansullivan07 Mar 28 at 14:48
    
This reminds me of something Henry Pollak once told me. I paraphrase: Indices are simple enough; indices of indices are more difficult for students to comprehend; the further down the rabbit hole you go, the more likely you are to lose your students. The idea of this iterative process for some reason is difficult for many. –  Andrew Sanfratello Mar 29 at 3:39
    
I would say indices also fit APOS-theory as they are "variables of variables". Meanwhile, I understand it's problematic to go down the rabbit hole. I far less understand why some people seem to grab down faster as physics allow... –  Anschewski Mar 29 at 7:49

The reason that similarities arise in sequences and series is that in discussing the convergence of series and other properties, you're actually working with a partial sum of a series, like "the first n terms".

These partial sums, however, are not themselves a series: they in fact form a sequence.

For instance, the partial sums of 1 + 2 + 3 + 4 + ... form the sequence 1, 3, 6, 10, 15. If such a sequence converges on a value, then the series converges.

The partial sums, and the sequence which they form, are not objects which are not interchangeable with the original series from whose fragments they are derived.

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