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Say I have some sequence $\{a_n\}$ with one subsequence $\{a_{n_i}\} \longrightarrow \infty$ and another $\{a_{n_j}\} \longrightarrow -\infty$. In other words, the lim sup $a_n = \infty$ and lim inf $a_n = -\infty.$

Because the sequence clearly does not converge, I am guessing I can call $\{a_n\}$ divergent. However, does $\{a_n\}$ diverge to $\infty$ and $-\infty$, or does it diverge to neither?

Just trying to make some sense of the definition of "divergence to infinity." My guess is that $\{a_n\}$ diverges, but does not diverge to either positive or negative infinity, since we can always find some element of the sequence greater than an arbitrary $M$ and another element less than $M$.

Many thanks.

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    Consider the sequence $ (-1)^n n$, it satisfies the conditions you mentioned at the beginning Is that what you want?2012-11-08
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    Yes! That sequence seems to fit my description. Any ideas regarding the divergence issue?2012-11-08
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    The last part is true as well and the sequence I just gave satisfies the last part2012-11-08
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    It diverges, full stop. *Diverges to* $\infty$ is just verbal shorhand for a particular kind of divergence behavior that a sequence of the kind that you’re talking about does not exhibit, and the same for *diverges to* $-\infty$.2012-11-08
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    So to say "diverges to infinity" does the following equality need to hold: lim inf = lim sup = $\infty$?2012-11-08
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    I agree with Brian M.Scott2012-11-08
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    @dirk5959: Yes, because you have to have $\lim_{n\to\infty}a_n=\infty$.2012-11-08

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We say a sequence diverges if it doesn't converge.

It is an abuse of terminology to say that the sequence "diverges to $+\infty$" or "diverges to $-\infty$", though people use it frequently.

What typically is meant by diverging to $+ \infty$ is the following:

$$\text{For any $M>0$, there exists $N \in \mathbb{N}$ such that for all $n > N$, we have $x_n > M$.}$$

Similarly, for diverging to $-\infty$.

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    Then it is not imprecise. And it is actually good to use that terminology since it gives more information about the sequence. A sequence that diverges to $+\infty$ is convergent in a compactification, property not satisfied by arbitrary divergent sequences. The concern raised in the comments above is that from the information given one cannot even tell if that sequence diverges to infinity.2018-03-23
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    @SphericalTriangle How is "a_k diverges to infinity" better than ""lim inf a_k = +infinity"?2018-03-23
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    @Acccumulation If you derived that conclusion from what I said you don't have good command of rules of inference. I have trouble talking to people that can't follow basic logic. Read there, if you don't understand, try harder.2018-03-23
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    @SphericalTriangle To say something is good is to imply that it's better than the alternatives, or at the very least not significantly worse. Your tone is rather rude.2018-03-23