Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

a_n=n*(-1)^n

Question

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

a_n=n*(-1)^n

roadjester · Answer

$$a _{n}=n (-1)^n$$

anonymous · Answer

It is not increasing or decreasing because of the -1

roadjester · Answer

Sure, but is it bounded?

anonymous · Answer

$\{-1,2,-3,4,-5,6,-7,8...\}$

kinggeorge · Answer

The absolute value of the sequence is just $a_n=n$ Is this bounded?

anonymous · Answer

surely does not look bounded to me

roadjester · Answer

And I'm taking the absolute value because why?

roadjester · Answer

I thought I was taking a limit?

kinggeorge · Answer

Taking the absolute value is just an easier way to see if it's bounded or not. Formally, it's not enough, but it allows you to see why it isn't bounded.

roadjester · Answer

But I thought the theorem was: $$\lim_{n \rightarrow \infty} \left| a_n \right|=0$$ then $$\lim_{n \rightarrow \infty} a_n=0$$

kinggeorge · Answer

That's correct. Like I said, my explanation is not good enough for a proof. It's merely an intuitive way to see the problem.

roadjester · Answer

So is it applicable to this situation since the abs is not 0? I mean, since the abs isn't 0, then wouldn't there have to be another way to determine whether it was bounded or not even though intuitively we can tell it isn't?

kinggeorge · Answer

I might show that the distance between two numbers in your sequence does not approach 0. $$\lim_{n \rightarrow \infty}a_{n+1}-a_n$$

kinggeorge · Answer

$$=\pm(2n+1)$$