Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Is it correct to say that the sequence $\left<(-1)^nn\right>$ tends to both $\infty$ and $-\infty$, since it eventually alternately approaches (oscillates between) $\infty$ and $-\infty$?

Or should I instead say that the sequence $\left<(-1)^nn\right>$ tends to neither $\infty$ and $-\infty$, since there is no single limit point (considering for our purposes that $\infty$ is a limit point)?

The latter sounds more technically correct, but it's so misleading though. Please advise.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Definition: A sequence $\{s_n\}$ tends to $\infty$ if for any value $x$, there exists an $N$, such that for all $n > N$, we have $s_n > x$. Similarly, a sequence $\{s_n\}$ tends to $-\infty$ if for any value $x$, there exists an $N$, such that for all $n > N$, we have $s_n < x$.

You can show from this definition, that a sequence cannot tend to 2 different values in $\mathbb{R} \cup \{-\infty, \infty\}$. Likewise, your sequence $\{ (-1)^n \cdot n\}$ doesn't tend to $\infty$ or $-\infty$.

share|improve this answer
    
+1, couldn't have said it any better. However, I would like to mention that $\left<(-1)^n\right>$ does have two limit points, namely $1$ and $-1$. Note that "tending to $\infty$" in particular means having no limit point. –  Jesko Hüttenhain Jan 9 '13 at 8:35
    
Thanks for the argument-from-definition reminder! I know I could simply elaborate my point that the sequence alternately approaches $\infty$ and $-\infty$ (by referring to its subsequences for example), but I still find it a tad misleading to claim that the sequence approaches neither $\infty$ nor $-\infty$, even though that is technically correct... –  Ryan Jan 9 '13 at 8:45
    
@JeskoHüttenhain I like the question "Show that a sequence has no limit points if and only if the sequence tends to $\infty$ or $-\infty$". It has ideas of compactness, which isn't officially introduced. –  Calvin Lin Jan 9 '13 at 8:50

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.