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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 $x_n \to \infty$ ?

3 Answers 3

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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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    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
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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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[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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    @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