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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 '14 at 23:42
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 '14 at 23:46
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 '14 at 2:04
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 '14 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 '14 at 13:38

10 Answers 10

up vote 29 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 '14 at 0:12
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 '14 at 0:12
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 '14 at 6:39
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 '14 at 14:31
@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 '14 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 '14 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 '14 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 '14 at 10:11
+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 '14 at 16:06
Sequences before series may not have caught on in the US, but in France, in the first math courses after high school (which are most of the time not called calculus by the way), sequences usually come before series, sometimes as much as a semester before. – Thomas Richard Apr 10 at 7:53

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 '14 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 '14 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 '14 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 '14 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 '14 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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Well, the first step to recovery is admitting that you have a problem. Just by acknowledging that students have this misconception, you're already well on your way to resolving it. Students are a lot less likely to make this mistake if you draw attention to it.

I'd offer 2 main explanations as to why this occurs in the first place:

  1. The words "sequence" and "series", in common parlance, can mean the same thing. Nobody would distinguish between the phrases "sequence of events" and "series of events". Because of this, students who are either struggling or half paying attention might assume you're using two words for the same concept.

As others have pointed out, you can have fun with this and point out the absurdity of saying things like "TV sequence", "World Sequence of Poker", etc. Noting that this is a misconception students have to overcome forces them to be more vigilant on exams whenever dealing with sequences or series.

  1. Series involve sequences in a number of confusing ways:

(i) A series is, colloquially, formed by adding the terms of a sequence together.

(ii) Formally, the definition still involves a sequence of partial sums. Even though students almost never fully understand this (they rely on their informal understanding of a series as a sum of infinitely many terms, however erroneous that may be), most instructors feel obligated to at least mention the formal definition, which can cause confusion if you aren't careful.

(iii) The convergence or divergence of a series $\sum{a_{n}}$ can be determined by properties of the sequence $a_n$ - for example, if $\displaystyle\lim_{n\to\infty}a_{n} \neq 0$, $\sum{a_{n}}$ diverges. This gets complicated by the fact that students confuse the limit (sum) of the series with the limit of the sequence on which the series is based. So you have to reassure students that a series can converge if its sum is nonzero, but not if the limit of $a_{n}$ is nonzero. Having to constantly refer to related but distinct mathematical objects and how they affect each other makes it all quite garbled.

Series are easily my favorite topic, and I taught a group with a 90% pass rate and 100% positive course evaluations - and I still struggled to get this distinction across to some students. So, good luck with that.

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Here are a couple more suggestions to drill in students' heads that sequence means progression and series means sum.

Consider a sequence of "episodes", each of which is a set of events. Episodes can be added by taking their union. The events that have happened so far as of an episode is the sum of it and all previous episodes. We call this sum a TV series. If your students aren't ready to accept a union as a type of sum, you could start with an "anyone can die" series, define an episode's value as the number of deaths in that episode, which gives the series as the total body count.

Consider a sequence of "ballgames", each of which has a value of 1 for a win or 0 for a loss. Then the World Series is a sum that approaches 4 for one of the teams.

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Sequence is just a function of the type $f:\mathbb{N} \to \mathbb{R}$. It is common to list the elements of this sequence as $$(a_1,a_2,a_3,\ldots,a_n)\,.$$ One example is the sequence of all even numbers: $(0,2,4,6,8,10,\ldots)$. However some sequences may be defined in a different form when there is no easy formula for expressing the terms, like the sequence of prime numbers $(2,3,5,7,11,13,\ldots)$, defined verbally.

Series means summation of terms, this is clear if we use sigma notation, in which the terms are defined by a law that resembles a sequence. For example: $$\sum_{i=1}^{n} i^2$$ is the sum of the sequence of squares $(1,4,9,16,25,\ldots,n^2)$. We can expand the RHS for the sake of clearness: $$\sum_{i=1}^{n} i^2=1^2+2^2+3^2+4^2+\cdots+n^2.$$

Often the series has a formula that depends only on the upper limit, so we can easily find the result without adding the terms, for the example above we know that $$\sum_{i=1}^{n} i^2=1^2+2^2+3^2+4^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$$

There is also the term infinite series, that is simply the limit $$\lim_{n\to\infty}\sum_{i=1}^{n} a_i\,,$$ from this many concepts arrises, but it is another discussion.

For the sake of didactic you can visualize a sequence as a stone path, where each stone has a number, when talking about sequences what matters is in what stone you are, like in the number 3 or more generally, $a_n$. On the same example, series is the path you travel to reach a determined stone, i.e., if you are supposed to go to the stone numbered 9 (from the origin), series is the sum of the stones you stepped, in this case: $(1+2+ 3+\ldots+8+9)$, or $(a_1+a_2+a_3+\ldots+a_8+a_9)$.

Hope I could help you more. I could, so I edited.

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I've given this a -1 because you seem to be answering the question "What is the difference between a sequence and a series?" rather than answering the question "How can I teach my students the difference between a sequence and a series?" I hope not to discourage you from future participation, but I also don't think this answers the question as asked. – Chris Cunningham Mar 31 at 12:46
Sorry, I missed the point, so I edited. – Brian Mayer Mar 31 at 19:43

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