36

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 ...


27

Since $\varphi$ is rather close to the conversion rate between miles and kilometers, one can use the Fibonacci numbers to convert: if $f_n$ is the distance in miles, then $f_{n+1}$ is (roughly) the distance in kilometers. You can use this to facilitate a discussion about, first of all, the convergence of these ratios $\frac{f_{n+1}}{f_n}\to\varphi$, and also ...


22

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 ...


16

I think the Fibonacci numbers should be mentioned in a first-year calculus class, though they should not be discussed at length. I wouldn't try to stress the real-world applications—these range from convoluted to dubious, and aren't really the reason that Fibonacci numbers are famous. Here are some good reasons to mention them: They are a famous ...


15

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 ...


11

Mathematics (People have already mentioned many of the great properties of this sequence at the time of answering. These are a couple of good ones that were missed.) Combinatorics The Fibonacci sequence is the number of strings of characters of length $n$, containing only a and b, where two a's cannot be consecutive. For $n=4$ there are $8$ strings for ...


10

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 ...


10

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 ...


9

My feeling is that the $\epsilon$-$\delta$ formulation is already pretty close to what one should think about limits; that is, the language can be hard to grasp at first, but the idea is very literally expressed in that language and can also be expressed in words. I find this more convincing using a physics measurement approach (which is slightly more ...


8

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 ...


8

My favorite counterexample is one about exchange of limits: $$a_{n, m} =\frac{n}{n+m}. $$ It is indeed pretty simple to see that $\lim_{n\to\infty} \lim_{m\to\infty}a_{n, m} \neq \lim_{m\to\infty} \lim_{n\to\infty}a_{n, m} $.


8

One idea: Try using function notation instead $$a(n+1)=a(n)+n.$$ It's more familiar than subscripts. Then ask for $a(m)$ possibly with the hint that $m=n+1$.


8

I have two intuitions to offer: A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for every neighborhood and every index $N$, there is an element $a_n$ with an index $n>N$ which is in this neighborhood). The $\limsup$ is the largest of these ...


7

If I were to cover the Fibonacci Sequence in introducing sequences to Calculus students, I would probably avoid some of the more obscure properties. I would focus on what early Calculus students should know: for example, how to prove that a sequence converges, and if it does, then how to find what it converges to. (Note also the places, not necessarily ...


6

Our math teacher came with the fun fact that the sum of the first 10 terms equal to 11 times the 7th term no matter what starting numbers you choose. Example: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11 * 13. The reason it works with any starting numbers is because you can rewrite it as: (a) + (b) + (a + b) + (a + 2b) + (2a + 3b) + (3a + 5b) + (5a ...


5

To explain the answer, I found this question while thinking about asking a similar question. In teaching both pre-calculus and calculus, this is an issue that seems to come up over and over again in my professional life—students need to be given some notion of limits so that the rest of calculus can be made to work, but the full rigor of $\varepsilon$-...


5

The relation with squares is pretty nice: $\sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1}$ The video link shows how suitable this concept might be for consumption by a general audience.


5

It depends on how Calculus is treated. At my university and others I've attended (US), the concept of the limit is usually treated early in the first Calculus class in order to talk about continuity and derivatives. The core idea of sequences is brushed over in the introductory courses (except in the advanced courses). Sometimes (and the department is ...


5

The linear function component is covered early(ish) in Algebra 1, and quadratic functions are covered towards the end of Algebra 1; so, the former by 7th/8th grade and the latter - if at all - by 8th grade. But, even for students in New York (where I teach) taking Geometry in 9th grade, more general polynomials may not be covered until Algebra 2 in 10th ...


4

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: The words "sequence" ...


4

Some number theoretic properties: For any number $N$ there exists a Fibonacci number divisible by $N$, which also means that for any $N$ there exists a Fibonacci number ending in $N$ zeroes. $F_n$ divides $F_m$ iff $n$ divides $m$ except for $F_2 =1$ which divides more stuff, this requires that you start with $F_0 =0$ and $F_1 = 1$. Following a similar ...


4

The students may need some support to imagine what the recurrence relation is telling them. Many may not realise that the sequence of a's is in some sense already there, and the recurrence relation tells them how to calculate one term from others no matter what part of the sequence they are looking at. My experience is that pictures often help students feel ...


4

The idea behind the limsup that you write is not simple and will not convey a concise intuition. The shortest description of the limsup of a sequence needs two steps: (1) The audience has to know what a limit point (also called accumulation point or cluster point) of a sequence is, and that a sequence can have many limit points, by examples. (2) Granting ...


3

The natural language description of a limit is usually "the number that the values in the sequence get closer and closer to". The phrase "closer and closer" strongly indicates a time element to sequences. That is, $n$ increasing is the same as time moving forwards. I imagine the values of the sequence as dots on a number line, with a new dot appearing every ...


3

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. ...


3

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 ...


3

Proofs That Count by Benjamin and Quinn has some interesting statements about Fibonacci numbers. They establish a correspondence between Fibonacci numbers and the number of ways to tile an $ n \times 1$ strip with tiles that are either $ 1 \times 1$ or $ 2 \times 1$, so every claim they present has a visual/combinatorial interpretation. A lot of the ...


3

My favorite counterexamples using sequences is, in fact, with regards to continuity. Many starting students have this notion in their head that to be continuous is essentially being able to draw the graph of a function without being able to pick up their pencils (pens, quills?, etc). Now, this is true for many functions that they are subject to in the usual ...


3

Every sequence $\langle u_n: n\in\mathbb{N}\rangle$ has a natural extension $\langle u_n: n\in{}^\ast\mathbb{N}\rangle$ where ${}^\ast\mathbb{N}$ are the hypernatural numbers (positive hyperintegers). The limsup of the sequence (when it is finite) is then exactly the sup of the shadows of $u_H$ for infinite indices $H$. So I would suggest explaning this as ...


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