# Is there a single word for the convergence / divergence of a sequence or sum?

In math, we sometimes talk about the parity of a number - that is, what Wikipedia succinctly describes as " an integer's inclusion in one of two categories: even or odd." Is there a similar word for convergence / divergence that would allow you to say "Find the X (convergence / divergence) of the sequence"?

• Asymptotic behaviour is close but gives you more detail than simply convergence or divergence. Feb 24, 2016 at 2:53
• You might say "find the limit." The reason there is unlikely to be a word is because while one can find a convergence value, one cannot find a divergence value. It doesn't make sense to say "find the divergence." Feb 24, 2016 at 4:20
• @Silenus I would've said "Determine if the sequence converges or diverges"; however, I chose what I did hoping it was possible to say something analogous to "Find the parity of the number." That doesn't necessitate a term, of course - especially when "divergent value" is nonsensical, as you said.
– llf
Feb 24, 2016 at 23:54
• I think the suggestion of @Lawrence (asymptotic behavior) deserves to be written up as an answer. Feb 24, 2016 at 23:57
• @Silenus Done :) . Feb 25, 2016 at 1:20

What you're referring to is the behaviour of the 'far-reaches' of sequence. In other words, the sequence's asymptotic behaviour.

Here's a literal rendering of the term asymptote as well as the more abstract application of the concept in asymptotic analysis:

A line that a curve approaches as you move further away from zero. - mathisfun.com
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. - wikipedia

Although analysing asymptotic behaviour provides information on whether a sequence converges or diverges, it typically goes beyond just convergence and divergence, looking for a function that essentially ignores complex behaviour that occurs in early parts of the sequence. It considers the question:

how is the system behaving "after a long time?" - Contemporary Calculus, section 4.6

If you want to restrict the consideration to just convergence and divergence, you can rephrase the question to "does the sequence converge?".

• +1. I agree with you mostly, except for your caveat regarding asymptotic behavior "going beyond just converge and divergence." If someone told me to "Determine the asymptotic behavior of the sequence", I would simply find out whether or not it converged, and if it did, I would tell them what it converged to. I don't think you need the caveat. Feb 25, 2016 at 1:28
• @Silenus That's an interesting point. I was hoping for that train of thought in the literature, but the ones I came across went all the way to finding the function. That is, the analysis can say a sequence doesn't converge, but if it says a sequence converges, there's an expectation to also say what it converges to. Feb 25, 2016 at 1:32
• Whoops, I misunderstood what you were getting at. I see the problem now and the reason for the caveat. Feb 25, 2016 at 1:38

Not a word, but a notation that can be unpacked easily. Two actually.

https://en.wikipedia.org/wiki/Big_O_notation

http://mathworld.wolfram.com/Little-ONotation.html

It is best known for characterizing the divergence of high growth rate behaviors such as the difficulty of solving a problem computationally as the size of the input grows. But it is equally suited to describing convergent series and is used that way in several specialties.