# Use of the word "convergent"

This question is for people who know some mathematics.

Is it correct to say

The sequence is convergent to 0.

Normally we say: The sequence converges to 0.

Without going into whether it is correct or not (I think it's acceptable), I can tell you why mathematicians use it. Sometimes, we want to emphasise the value of the limit. In such cases, we say "This sequence converges to 0" or "The limit of this sequence is 0". But sometimes, when we're more concerned with the analysis of the sequence (whether it is a convergent sequence, or divergent, or what), we want to emphasise the fact that it converges, rather than its limit. In such cases, instead of saying

This sequence is convergent, and its limit is 0"

or

This sequence is convergent, and it converges to 0"

we (or some mathematicians) use a shortcut and say, equivalently,

"This sequence is convergent to 0."

• …or, to get the emphasis right, add parentheses: "This sequence is convergent (to 0)." Jan 15, 2011 at 17:55

We usually say

The sequence converges to 0,

but it is also correct to say

The sequence is convergent to 0.

Having said that, we know that language usage in technical areas is different from that in everyday life. For example, we say the triangles are incongruent to mean they are not congruent. However incongruent is not a standard word in everyday use, and the opposite of congruent in the everyday sense is incongruous, which would not be used to describe triangles.

It all boils down to accepted conventions in technical usage then. Grammatically, convergent to 0 is an adjectival phrase, with the adjective convergent modified by to 0.

I believe it is technically correct, but stylistically I prefer the phrase "convergent toward 0".

• Really? This sounds odd to me (native British speaker, having studied mathematics at degree level); I would definitely prefer "convergent to" as suggested in the question. I think this is because "convergent" already contains the meaning of "moving towards a destination", so specifying that again with "toward(s)" is unnecessary and almost sounds like repetition. Jan 24, 2011 at 14:41

I think it is a bit sloppy to say 'the sequence is convergent to 0'. What if you said 'the sequence is not convergent to 0'? Does that mean it is not convergent?

• Suppose we say "Tom was not bitten by a wolf." Does that mean that he wasn't bitten, or that he was bitten but not by a wolf? Normal English doesn't clarify this, except maybe by context. Does that mean we shouldn't use the sentence "Tom was bitten by a wolf"? Why should we require that mathematical English clarifies a similar hypothetical negation? Apr 13, 2011 at 21:13