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I lost a point on my test because I put after all my math work, "converges at $0$" not "converges to $0$". Should I argue with my teacher about this or is there a reason why I am wrong?

Stella Biderman
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Travis Delly
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    Depends on the context. Both terms are used. See here. The term "converges at" refers to which value of some other parameter which makes the series converge, while the term "converges to" refers to the value the series converges to. – S.C.B. Mar 01 '17 at 03:44
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    Not a very compelling thing to argue about if you spell it "grammer." On the one hand, it seems a bit pedantic for the teacher to deduct a point. On the other hand, yes, the phrases can have slightly different meanings. In more advanced math you can say a series of functions converges at $x=1$ to a certain value, say. For a sequence of numbers rather than functions, "to" is idiomatic. – symplectomorphic Mar 01 '17 at 03:45
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    Common usage is to say that, for example, the series $\sum \frac{x^n}{n!}$ converges *to* $e$ *at* $x=1$. So, yes, there is an important distinction between the two. – dxiv Mar 01 '17 at 03:50
  • What is it that converges in this problem? Is it a sequence? A series? A mention of the course where this occurred might be helpful. – hardmath Mar 01 '17 at 03:53
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    GrammAR. ${}{}$ – Mariano Suárez-Álvarez Mar 01 '17 at 03:56
  • @Anonymous I think I clicked "edit" before you did, but then hit "save" after you did, which functionally causes my edits to reverse yours. I'll reinstate those changes. – Stella Biderman Mar 01 '17 at 04:01

2 Answers2

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If $\lim_{x \to a} f(x) = L$ then $f$ converges to $L$ at $a$.

addendum

The following is an exerpt from talkenglish.com

At:
   Used to indicate a place:
      There is a party at the club house.
      There were hundreds of people at the park.
      We saw a baseball game at the stadium.


To
   Used to indicate the place, person, or thing that someone
   or something moves toward, or the direction of something:
        I am heading to the entrance of the building.

  Used to indicate a limit or an ending point:
       The snow was piled up to the roof.
       The stock prices rose up to 100 dollars.
Steven Alexis Gregory
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    To be honest, if you said "1, 1/2, 1/4, 1/8, ... converges at 0", there's nothing else it could mean than converging to 0. It's not necessarily referring to something else. – user541686 Mar 01 '17 at 07:38
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    @Mehrdad If you said "1, 1/2, 1/4, 1/8, ... converges 0" most all readers would still parse it right. Problem is that such constructs often become more complicated, and then such ambiguity in the wording hurts. Which is why it's advised to practice the appropriate, good wording to begin with. – dxiv Mar 01 '17 at 08:09
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    @dxiv: That wasn't my point. My point was only that the distinction in the answer need not exist. That said, the difference between your version and mine is that yours is actually using invalid grammar whereas mine is actually both correct in terms of grammar and also in terms of the only logical meaning it could convey... the worst you can say about it is that it's not using the conventional terminology. So, no. – user541686 Mar 01 '17 at 08:14
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    @Mehrdad I see your point, appreciate it, and don't mean to argue that. My comment was in the context where the OP wrote about the assessment being on a "test". Tests are usually meant to enforce "best practices", and being unambiguous on possible *at* vs. *to* confusion could well be part of that. – dxiv Mar 01 '17 at 08:25
  • @Mehrdad - Either way, thank you for making me want to look up the proper use of "at" and "to". – Steven Alexis Gregory Mar 01 '17 at 08:57
  • @user541686 I know this is a lot later but I was looking over my old answers and this comment just occurred to me. If a sequence converges, then it converges to a limit at infinity. – Steven Alexis Gregory Jul 15 '20 at 22:42
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"at" and "to" mean importantly different things here. When you have a sequence, "at" talks about the domain, while "to" refers to the range. If you tell me a sequence coverages at $0$, my immediate thought is that the series evaluated at $0$ converges. However, if you tell me it converges to zero, my immediate thought is that its limiting value is $0$.

Here are some sentences that highlight the distinction. See which of the two words makes more sense in the sentence:

"The value of the function [at/to] $x=0$ is..."

"The function $f:x\to x^2$ maps [at/to] $[0,\infty)$"

Stella Biderman
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  • Small grammar thing: “it’s” → “its” at the end of the first paragraph. – Ry- Mar 01 '17 at 06:25