Is it meaningful that sequence or expression has limit $\infty$?

Question

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?

Answer 1

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 $\Bbb N\cup\{\infty\}$.

Answer 2

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.

Answer 3

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)