11
votes
Accepted
Motivating example for sequences, sums and limits in high school
This application is known as "gross-up" in accounting.
You run the finances for a small business. The boss would like to give an employee a \$100 bonus for their hard work. However, the ...
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11
votes
Accepted
What is the intuition behind the limit superior?
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 ...
- 2,992
9
votes
Is this motivation for the concept of a limit a good one?
The concept of a limit has nothing to do with the order on $\mathbb{R}$. The standard definition of a limit of a sequence generalizes, almost verbatim, to sequences with values in $\mathbb{R}^n$, ...
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8
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Substitution in recurrence relations
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 $m=n+1$.
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7
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Is this motivation for the concept of a limit a good one?
I don't like it. Had a hard time following it. Just tuned out. Yes, I'm not a Ph.D. in math. But neither will be the target students. You should have won me over. You didn't. I have the IQ to ...
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6
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Motivating example for sequences, sums and limits in high school
I'm not sure that starting with an applied motivation (derivation, word problem) is the best way to introduce this topic. Look at how your experiment failed. This is because "word problems are ...
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5
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How can I teach my students the difference between a sequence and a series?
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 ...
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How can I teach my students the difference between a sequence and a series?
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 ...
5
votes
Teaching limits of sequences before limits of functions in Calculus?
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 ...
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5
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Accepted
In what grade do kids (New York, US) learn common differences?
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 ...
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5
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What is the intuition behind the limit superior?
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 ...
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4
votes
Substitution in recurrence relations
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 ...
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3
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Is this motivation for the concept of a limit a good one?
Here is a simpler definition of limits which works better for monotone sequences, and has value in motivating further investigation of the concept of limits:
If a sequence $x_1, x_2, ...$ is ...
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3
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Is this motivation for the concept of a limit a good one?
While monotone behavior is important in analysis (such as the monotone convergence theorem in measure theory),
I think you should forget the emphasis on monotone behavior and just be honest with the ...
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3
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Is this motivation for the concept of a limit a good one?
I think the answer depends on the meta question: why are students learning the definition of a limit?
Here are some reasons I can think of:
Because students are taking an intro-to-proofs class and ...
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3
votes
Motivating example for sequences, sums and limits in high school
There are lots of good physics examples involving equilibrium. For example, you can set up a pendulum and show how the amplitude forms a sequence that decays exponentially toward zero, or describe ...
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3
votes
Motivating example for sequences, sums and limits in high school
The common puzzle of giving a few terms and asking for the next are examples of (generating) sequences by some particular rule.
A series is just a sequence, summed together. Ask e.g. for the sum $1 + ...
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3
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In what grade do kids (New York, US) learn common differences?
This is a topic I could imagine not being adequately covered in all U.S. schools, although (as Dave L. Renfro pointed out in a comment), it is listed in the Common Core Mathematics standards under ...
- 24.4k
3
votes
What is the intuition behind the limit superior?
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). ...
- 2,102
3
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How can I introduce a speech about the Fibonacci sequence creativiely?
ADVICE:
When you have a question about a speech, you should give us more information to help you. Especially how long is the speech (it affects how much intro you do). But also the audience. And ...
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Teaching limits of sequences before limits of functions in Calculus?
When you look at the formal definitions of several concepts in real analysis you see that those concepts are rooted in the limit of sequences (or better to say, these concepts can be defined in a way ...
- 2,021
2
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What is the intuition behind the limit superior?
I would probably avoid the phrase "at infinity", since it's kind of vague and I'm not sure if it'd help someone who doesn't already understand the concept. Instead, I would talk more explicitly about ...
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Motivation for Fibonacci: Bees
After looking at Tony Jacobs argument, here is an argument of my own.
Start with a single male. At each generation, let $m_n$ and $f_n$ be the total number of males and females respectively, and let ...
- 1,078
2
votes
Accepted
Motivation for Fibonacci: Bees
You can see it by breaking the numbers $s_n$ into parts: $s_n=f_n+m_n$, which represent the number of female and male bees, respectively, at each level of the family tree.
To find $f_{n+1}$, we note ...
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What is the intuition behind the limit superior?
For me, this picture from wikipedia helped the most in understanding limsup and liminf
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Is this motivation for the concept of a limit a good one?
Two points to think about:
It might be more intuitive to replace "from some $N$ onward" by "for all $n$ except finitely many".
I often present the notion of a limit through a game:...
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1
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What is the intuition behind the limit superior?
Limit supremum (of a sequence of numbers)
I will try to give intuition only using words, without using mathematical symbols.
What is the smallest number which is greater than infinitely many members ...
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Any metaphors/intuitions for a limit of a sequence?
Here is my "unfinished" attempt that I used in one of my classes last year. Generally speaking, it is based on the idea of "proof-generated definitions" introduced by Lakatos. As such, the question is ...
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How can I teach my students the difference between a sequence and a series?
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 ...
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