# Series vs. sequence

I got dinged on a paper once because I mixed up the notions of series and sequence. I've never bothered to really clarify the distinction. Can anyone tell me what the difference is? (We're talking math here.)

Tom Mattson
Staff Emeritus
Gold Member
A sequence is just that: a sequence of terms that follows a rule of some sort. A series is a summed sequence.

Ok, so <1, 10, 19, 28, ..., 9(n-1)+1, ...> is a sequence. Since it's divergent, I don't see how it can be summed. Unless you just mean including infinity as a final element.

Can you give me examples of sequences that are series and some that aren't? Maybe I'm just not seeing the point of the distinction. It could also be that I just got a contentious reviewer.

Tom Mattson
Staff Emeritus
Gold Member
Originally posted by drnihili
Ok, so <1, 10, 19, 28, ..., 9(n-1)+1, ...> is a sequence. Since it's divergent, I don't see how it can be summed. Unless you just mean including infinity as a final element.
A series does not have to be an infinite series, so it can be summed if you impose a finite upper bound on n. But even if you do let n-->infinity, you still have a series, you just have a divergent one as you noted.

Can you give me examples of sequences that are series and some that aren't? Maybe I'm just not seeing the point of the distinction. It could also be that I just got a contentious reviewer.
I think you are not seeing the distinction.

A sequence is a set of numbers.
A series is sum of a sequence.

Example,

The set {1/2,1/4,1/8,...1/2n|n-->infinity} is a sequence of numbers prescribed according to the rule 1/2n.

The sum &Sigma;n(1/2)n for n=1 to infinity is a series that is equal to 1.

Ah, ok I got it. And yes, it was a contentious reviewer.

HallsofIvy
Homework Helper
A reviewer? You were attempting to publish a paper dealing with sequences and/or series and don't even know what they are? Sounds to me like a GOOD reviewer.

Originally posted by HallsofIvy
A reviewer? You were attempting to publish a paper dealing with sequences and/or series and don't even know what they are? Sounds to me like a GOOD reviewer.
No, I was attempting to publish a paper not dealing directly with the distinction between sequences and series. I used the term "series" to describe the number of laps a runner runs throughout a race, (i.e. 1, 2, 3, 4, ...). The reviewer stated that I was misusing terminology and that hence my conclusion didn't follow. From Tom's description above, it's not clear that I was misusing terminology as my primary concern was with the ongoing total of laps, though the transition between natural language and math is sometimes less than determinate.

Furthermore, this was a philosophy journal, not a math journal, and I was using the words in a non-technical context. Most importantly, the series/sequence distinction makes no difference to the conclusion I was attempting to draw. The conclusion is a bit heretical, so the reviewer apparently took the terminological dispute as a reason to reject a conclusion s/he didn't agree with. No other reason was offered btw.

I'm more than happy to be corrected on terminology, and I don't mind having papers rejected on good grounds. But when a reviewer rejects a paper based on a terminological distinction that is innappropriate to the paper and which would not bear on the conclusion even if it were appropriate, then that reviewer is just being contentious.