How can we teach students the difference between sequences and series?

[Meden Agan]

Sequences and series are related concepts, but they differ extremely from one another. I believe that students in integral calculus often confuse them. Part of the problem is that:

1. Sequences are usually taught only briefly before moving on to 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 one use to teach students to distinguish between sequences and series? Specifically, strategies that address the above concerns. Answers are greatly appreciated if they include appropriate references.

 

[PeroK]

If a student struggles with this, then mathematics isn't for them. There are a lot harder concepts than this.

 

[Steve4Physics]

What methods can one use to teach students to distinguish between sequences and series?

The difference between sequences and series is trivial. Are you sure you’ve asked what you intended?

But there could be casual misuse of terminology. Typically a student might use the word 'series' when they mean 'sequence'. (A bit like a physics student using 'velocity' when they mean 'speed'.)

 

[mathwonk]

Perhaps the answer is contained in your first point. I.e. perhaps one should teach sequences first and thoroughly, and only treat series later. This is done in Richard Courant's excellent calculus book vol.I, where sequences are given importance from page 27, and, after Taylor series, general series are only treated starting from page 366. One can also point out that sequences make sense using only a notion of distance, and the elements of the sequence need not even be capable of being added, such as the points on a sphere. So series are a special case of sequences.

 

[Herman Trivilino]

Sequences and series are related concepts, but they differ extremely from one another. I believe that students in integral calculus often confuse them.

What are you encountering that leads you to this conclusion? Is it nothing more than a vocabulary issue? In other words they haven't committed to memory the definitions of each so they know lots of stuff about them except for their names.

Most of these types of difficulties are the instructor's fault. You have to make their grade suffer for failure to learn.

 

[Muu9]

I think the previous idea of emphasizing or at least covering examples of sequences where addition makes no sense, like sequences of points on a sphere, is a great idea.

 

[symbolipoint]

If a student struggles with this, then mathematics isn't for them. There are a lot harder concepts than this.

I was confused about the difference for at least a couple of years. Then I looked carefully on the published written instruction, thought very carefully, and understood. Some students figure it out fast. Some need to study again to figure out.

 

[symbolipoint]

What are you encountering that leads you to this conclusion? Is it nothing more than a vocabulary issue? In other words they haven't committed to memory the definitions of each so they know lots of stuff about them except for their names.

Most of these types of difficulties are the instructor's fault. You have to make their grade suffer for failure to learn.

Debatable. Maybe yes. Maybe not. Ask each student. When both sequences and series are too new for student, he/she/they/it can get confused.... for a short time only.

 

[symbolipoint]

As best I can recall , the intermediate algebra textbooks do a great job in the transition from sequences, to series, and correspondingly handling and explaining the vocabulary.