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Questions tagged [sequences]

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29
votes
11answers
6k views

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

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: Sequences are usually ...
21
votes
11answers
8k views

What interesting properties of the Fibonacci sequence can I share when introducing sequences?

The Fibonacci numbers are one of the first sequences given as examples of sequences in many calculus textbooks as they have a definition that does not obviously have a closed form and they have many ...
17
votes
4answers
575 views

What are your favorite instructional counterexamples on sequences?

In this article, I give counterexamples regarding real sequences. And in that one some others. In particular counterexamples answering questions like: "If for all $p \in \mathbb{N}$ $\lim\limits_{n \...
17
votes
4answers
896 views

Teaching limits of sequences before limits of functions in Calculus?

Most Calculus courses/textbooks I have seen teach the different topics in that order: limits of functions, continuity, differentiability, integration. Then, depending on the teacher/textbook's ...
10
votes
2answers
175 views

Substitution in recurrence relations

Let's say we study a reccurence relation such as $$a_{n+1}=a_{n}+n, n \geq 1$$ I find many students are having difficulties when you ask them to find $$a_{n}$$ in terms of $$a_{n-1}$$ where they just ...
8
votes
6answers
345 views

Any metaphors/intuitions for a limit of a sequence?

I'm writing (together with a colleague) a minicourse on mathematical analysis (currently we want to cover the Weierstrass theorem on functions on compact intervals, so the aim is to present only the ...
6
votes
4answers
273 views

Are convergent and divergent sequences a prerequesite for calculus or are they a part of it?

Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: http://www.stitz-zeager.com/szprecalculus07042013.pdf However, it does not cover ...
5
votes
4answers
687 views

What is the intuition behind the limit superior?

I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
5
votes
2answers
156 views

In what grade do kids (New York, US) learn common differences?

I'm teaching an after school workshop for a few 7th graders. I was having them try to predict the next item in a complicated sequence. After some failed attempts, one of the kids started analyzing the ...
5
votes
2answers
172 views

Motivation for Fibonacci: Bees

I want to talk about the Fibonacci sequence in my Linear Algebra class. So I tried to look online for examples where the sequence appears naturally. One of the most often mentioned is that of the ...
1
vote
1answer
141 views

How can I introduce a speech about the Fibonacci sequence creativiely?

I am a high school senior student. Soon, I am giving a presentation about the Fibonacci sequence and I am searching for a creative way to start my speech. I was wondering whether someone in this ...