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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 $x_n \to \infty$ ?

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3 Answers 3

up vote 2 down vote accepted

[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 $f(x) \to -\infty$.

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alright thanks :) i provided you with a link of the notes: rutherglen.science.mq.edu.au/wchen/lnfycfolder/fyc19.pdf on page 4 Example 19.1.15 –  student101 Sep 26 '12 at 14:11
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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 lectures –  student101 Sep 26 '12 at 14:13
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@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. –  Thomas Sep 26 '12 at 14:15
    
Thank you again Thomas –  student101 Sep 26 '12 at 14:21
    
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 $x_n\to\infty$. –  Thomas Andrews Sep 26 '12 at 14:24
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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, $|x_n|\to+\infty$.

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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.

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so it either a case of 'absolute convergence' or a case of a different definition; thanks –  student101 Sep 26 '12 at 14:35
    
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... –  Thomas Andrews Sep 26 '12 at 14:43
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