Questions tagged [sequences]

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33 votes
11 answers

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 ...
Brian Rushton's user avatar
22 votes
11 answers

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 ...
Brian Rushton's user avatar
19 votes
4 answers

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 ...
Taladris's user avatar
  • 1,367
18 votes
4 answers

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 \...'s user avatar
10 votes
2 answers

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 ...
amarius8312's user avatar
8 votes
6 answers

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 ...
Stephan Kulla's user avatar
8 votes
6 answers

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 ...
mbork's user avatar
  • 1,299
7 votes
6 answers

Motivating example for sequences, sums and limits in high school

I currently work as a substitute teacher at a local high school and the topic in one of the classes is sequences, series and limits. Because I always disliked learning about a topic without having ...
Aaron Daniel's user avatar
6 votes
4 answers

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: However, it does not cover ...
Anton's user avatar
  • 143
5 votes
2 answers

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 ...
j0equ1nn's user avatar
  • 437
5 votes
2 answers

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 ...
Martin Argerami's user avatar
3 votes
6 answers

Is this motivation for the concept of a limit a good one?

tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
Asaf Shachar's user avatar
1 vote
1 answer

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 ...
Merderzo's user avatar