3
$\begingroup$

My notes state:

$x_n= (-1)^nn$

satisfies: $x_n \to \infty $ as $n \to \infty$

however to me $x_n$ appears to alternate between positive and negative values of infinitismally large magnitude,

so why does my textbook describe its limit as $x_n \to \infty$ ?

3 Answers 3

2

It really depends on definitions.

There are two ways to "compactify" the real line. One is to add two points, $+\infty$ and $-\infty$. The other way is to add one point at infinity, $\infty$. The latter is, in a sense, more canonical - every space has a "one-point" compactification, while only and "ordered space" has a two-point compactification.

Alternatively, you might say that $x_n\to\infty$ if $|x_n|\to +\infty$.

I really dislike the usage of $\infty$ to mean $+\infty$ for this reason, so I agree with the book's usage. It might be confusing that $+\infty$ is different from $\infty$, but infinity is confusing.

  • 0
    so it either a case of 'absolute convergence' or a case of a different definition; thanks2012-09-26
  • 0
    No, the term "absolute convergence" is really only used for series and integrals, not for absolute values of a sequence. Instead, think of adding two infinities as making the real line "like" an interval, $[-1,1]$, while adding a single point at infinity is "like" making the real line into a circle. There is a "topological" sense in which this is exactly what is happening...2012-09-26
4

The sequence is $-1,2,-3,4,-5,\ldots$. If one is working in the extended reals, one may talk about $-\infty$ and $+\infty$. In this case, the sequence goes to neither. However, $|x_n|\to+\infty$.

2

[Your textbook might say that $x_n \to \infty$ and $x_n \to \infty$, but you are right. The sequence is divergent and the $x_n$'s have alternating sign.] In general then I would not write as your book has done.

However, you might look at the books definition of what $... \to \infty$ means. I guess that the book might include in this notation the case where the limit is as in this sequence.


Edit: After looking at the authors webpage and looking at chapter 6 page 9, the authors definition of $f(x) \to \infty$ as $x\to a$ is that $\lvert f(x)\lvert$ can be made as large as possible. Hence for all this to be consistent, there doesn't appear to be a mistake in chapter 19 on sequences.

Note that sometimes we might like to be able to distinguish between $f(x) \to \infty$ and $f(x) \to -\infty$.

  • 0
    alright thanks :) i provided you with a link of the notes: http://rutherglen.science.mq.edu.au/wchen/lnfycfolder/fyc19.pdf on page 4 Example 19.1.152012-09-26
  • 1
    its just i thought it would be extremely unlikely the notes could be wrong since they have been used for the last 10 years in lectures2012-09-26
  • 1
    @student101: Sure, I understand. But it does indeed look like there is a typo in that example. I quickly looked through the first 4 pages, and I don't see any definition of what the author means when he writes $x_n \to \infty$. As mentioned in the answer, if there is no typo, I wouldn't say that the authors notation is standard.2012-09-26
  • 0
    It partly depends on the definition of $x_n\to\infty$. There essentially two ways to "complete" the real line - one is to add two points, $+\infty$ and $-\infty$ the other is to add just one point,$\infty$. In the latter case, it is correct to say $x_n\to\infty$.2012-09-26
  • 0
    @ThomasAndrews: Just as a mentioned in my answer. Also, the notes are called "First Year Calculus", so I doubt that the students deal with "completions" of the real line.2012-09-26
  • 0
    Yeah, the higher math was for you. My point is, there can be a notion of $\infty$ that is different than $+\infty$ and $-\infty$ in limits, and that possibly the lecturer is making this distinction.2012-09-26
  • 0
    @ThomasAndrews: Ahh ok. Thanks. I edited my answer by the way and found that there isn't a mistake in the notes.2012-09-26