How can I teach my students the difference between a sequence and a series?

Question

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.

Answer 1

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.

Addendum: 11/6/2019: As the comments below describe, it surely is important for students to recognize that a term (like "series") may have a specific meaning in the context of mathematics that differs from its meaning in more colloquial contexts. So, I do not actually exclusively use the term "infinite sums" and avoid saying "infinite series". Instead, when this topic first arises in a calculus course, I show this very post to my students!

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.

2. 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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

Answer 11

First answer: Isn't sequence like a list of numbers that appear as according to function? And series is more like the addition of the terms of the sequence. I guess There convergence properties differ from one another. Can it not be thought like a function happening from N to R (sequence) and then the sum of the resulting numbers up to 'n' or infinity the series?

But can the convergence in sequence have any influence on the convergence of a series?

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Second answer: If I take the function from N to R as 1/n, its sequence (function) will have the numbers 1, 1/2, 1/3 etc listed in its range or will be the 'terms' . Now this particular sequence will converge to 0 b/c I will find some abscilla which is grater then the difference of any term and 0. However if I talk bout series(join, addition, summation) , it will be the sum of all the terms of the sequence I mentioned above and that same may not necessarily be smaller than abscilla which is why this series will not converge to a particular value.

I hope this is the difference and hope it's elaborate enough.