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Some students write, e.g., "$\sum(1/n^2)$ converges at $\pi^2/6$", where I would write "converges to".

Are there regions of the English-speaking world where it is standard to say "converges at"? Or should this be considered a mistake made by non-native speakers of English?

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I would expect it to be used in contexts such as "$\sum_k \frac{x^k}{k!}$ converges at 2", meaning that if we set $x=2$ then the resulting series converges. –  Henning Makholm Oct 27 '11 at 17:22

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up vote 17 down vote accepted

This is an error. Worse, there is a correct use for "converges at" which is not this. The following sentence shows a correct use of "converges at".

The sum $\sum_{n>0} \frac{x^n}{n^2}$ converges at $x=1$.

In other words, "converges at" refers to which value of some other parameter makes the sum converge, not to the value of the sum.

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@Asaf, better square that $n$ in the denominator. –  Gerry Myerson Oct 27 '11 at 23:40
@Gerry, you are much correct sir. The trouble of writing a comment whilst talking on the phone, I guess. –  Asaf Karagila Oct 27 '11 at 23:41
Strictly speaking, $f(x)=\sum_{n=1}^\infty\dfrac{1}{n^2}$ converges at $x=\dfrac{\pi^2}{6}$, it also converges to $\dfrac{\pi^2}{6}$ at that point... :-) –  Asaf Karagila Oct 27 '11 at 23:44
Since nobody has come forward to say that "converges at" is correct English (somewhere) for the situation I described above, I will accept that it should indeed be considered an error (as I suspected). –  idmercer Oct 29 '11 at 21:16
I am satisfied that the people who have replied are correct. But today, I am grading a test, and I notice an ENORMOUS number of students using "converges at" in the incorrect way I describe above. Most of those (I am guessing from their names and in some cases conversing with them) are native English speakers. Where are they getting it? –  idmercer Nov 1 '11 at 21:59

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