Question

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

a_n=n*(-1)^n

Answer 1

\[a _{n}=n (-1)^n\]

Answer 2

It is not increasing or decreasing because of the -1

Answer 3

Sure, but is it bounded?

Answer 4

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

Answer 5

The absolute value of the sequence is just \(a_n=n\) Is this bounded?

Answer 6

surely does not look bounded to me

Answer 7

And I'm taking the absolute value because why?

Answer 8

I thought I was taking a limit?

Answer 9

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.

Answer 10

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

Answer 11

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.

Answer 12

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?

Answer 13

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

Answer 14

\[=\pm(2n+1)\]