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I have recently started my studies in university (mathematics), and we're now studying sequences. I was surprised when the professor wrote that:

$$\lim_{n \to \infty} a_n=\lim_{n \to \infty}(-3)^n=\infty$$

It was stated $\infty$ means "the sequence takes larger and larger values in absolute value", while $+\infty$ means "the sequence takes larger and larger positive values".

I think that it is not really appropriate to say the sequence has limit infinity but that it has no limit, and rather diverges.

Similarily, I was thinking that the notation

$$\lim_{n \to \infty}a_n$$

is rather an abuse of notation, since as discussed in other questions, $\infty \notin \mathbb N$, and it'd be better to write

$$\lim_{n \in \mathbb N}a_n$$

and to read it as "limit of $a_n$ over $\mathbb N$".

Why would we insist on this auxiliary notion? Do you have any other example where it is taught like this?

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The notation is intuitive, even if it is an abuse. Also, when you say that the limit of a sequence is $(+)\infty$, you convey information about how the sequence fails to converge, which is more than just saying that the sequence diverges. –  Brett Frankel Apr 17 '12 at 21:14
@Andre: Even in a one-point compactification of $\mathbb R$? –  Asaf Karagila Apr 17 '12 at 21:16
@AsafKaragila: I am sure that was not the context! –  André Nicolas Apr 17 '12 at 21:18
@Andre One should not be so sure that the one point compactification is not intended. For example, my high school calculus teacher introduced me to analogous ideas (in simpler language). –  Bill Dubuque Apr 17 '12 at 21:51
Since your teacher clearly explained what was intended by the notation, I don't see what the problem is. Also I find the seemingly derisive ad hominem tone of this question a bit off-putting. –  Grumpy Parsnip Apr 18 '12 at 1:18

3 Answers 3

up vote 9 down vote accepted

I have never encountered that convention; in my experience $\lim\limits_{n\to\infty}a_n=\infty$ is synonymous with $\lim\limits_{n\to\infty}a_n=+\infty$. I don’t think that it’s a particularly useful convention, either. I am much more likely to want to know that the sequence diverges to $(+)\infty$ or to $-\infty$ than to know that the corresponding sequence of absolute value diverges to $\infty$. Moreover, in the extended real numbers the statements $\lim\limits_{n\to\infty}a_n=(+)\infty$ and $\lim\limits_{n\to\infty}a_n=-\infty$ are meaningful statements of convergence, but $\langle (-3)^n:n\in\Bbb N\rangle$ still doesn’t converge.

There is no abuse of notation in $\lim\limits_{n\to\infty}a_n$: it’s defined to have the usual meaning. And it’s a nice, intuitive notation that even makes sense when $\Bbb N\cup\{\infty\}$ is viewed as the one-point compactification of $\Bbb N$ and the limit of a sequence as the value that can be assigned to $\infty$ to make the extended sequence a continuous function on $\Bbb N\cup\{\infty\}$.

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Can you tell me a little more about the one point campactification of $\mathbb N$? –  Pedro Tamaroff Apr 17 '12 at 21:28
@Peter: The points of $\Bbb N$ are isolated, and nbhds of $\infty$ are sets of the form $\{\infty\}\cup(\Bbb N\setminus F)$, where $F$ is any finite subset of $\Bbb N$. It’s compact, and $\infty$ is the limit of the sequence $\langle n:n\in \Bbb N\rangle$. –  Brian M. Scott Apr 17 '12 at 21:31
What does "nbhds" stand for? –  Pedro Tamaroff Apr 17 '12 at 21:32
@Peter: Neighborhoods; the open sets that contain the point $\infty$. –  Brian M. Scott Apr 17 '12 at 21:33
Do you have any reference or link on that theory? I guess it concerns me. –  Pedro Tamaroff Apr 17 '12 at 21:34

The notion of "unsigned infinity" is very common when dealing with complex numbers, since they lack such an obvious notion of sign. This idea has a famous name attached. There is no inherent reason why you couldn't adopt a similar convention for real numbers, just that it isn't usually done.

This is a special case of the idea of a one point compactification.

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I can't relate this to what I'm asking. Could you explain a little more what you're saying? I'm not very educated in topology so I could use some pedagogy in your answer. Sorry! –  Pedro Tamaroff Apr 18 '12 at 0:14
The one point compactification is defined by adding a single extra point called $\infty$ to your space, and every sequence that eventually leaves every bounded set is defined to have limit $\infty$. In the case of the real numbers, the two ends, positive and negative join in a single point at $\infty$. For the complex numbers, the plane curls up to a sphere with a point at the north pole added. –  Grumpy Parsnip Apr 18 '12 at 1:13
@Jim Thanks. I get it now. –  Pedro Tamaroff Apr 18 '12 at 1:31

Regarding the limit: You are correct and I would say that it is your professors that are mistaken.

The limit of $|a_n|$ exist and equal to infinity but the limit of $a_n$ does not exist as it changes from very large numbers to their negation thus approaches 'nothing' (not a real number not +infinity nor -infinity)

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