Question

Construct a real sequence with the given property. Justify your answer:
The only subsequential limits are 2, -3, infinity

Answer 1

The only example I have is:
\[a_{n} = \frac{ 1 }{ n }+(-1)^{n}\] which has subsequential limits of -1 and 1. So, I understand that they chose even numbers and odd numbers to come up with -1 and 1 and then showed how other subsequential limits can't exist. But is this the same idea I'd need to use to construct a sequence for my problem? This can't just be a guess and check process to see if you get the limits you want, can it?

Answer 2

Hint: (2-3)/2 = -1/2

Answer 3

I'm not sure I understand that hint, I'm sorry x_x I don't see what I could do with that which would give me all 3 subsequential limits. Maybe it is a good hint, but this is brand new to me and I feel like there's a lot I don't know about the concept.

Answer 4

In your example, we have a function that approaches zero, 1/n and one that oscillates between 1 and -1.

I thought it might be fun to use the same idea to construct a function that approaches -1/2 and another that oscillates between +5/2 and -5/2.

Maybe: \(a_{n} = -\dfrac{n}{2*n} + \dfrac{5}{2}(-1)^{n}\)

It's a little awkward and does not provide the \(\infty\) requirement, but it might be a place to start.

Answer 5

I actually used a function just like that. I had:
\[a_{n} = \frac{ 1 }{ 2 } + (-1)^{n}\frac{ 5 }{ 2 }\] Only problem is as you mentioned, getting infinity. It seemed like, at least from the one example I had, that you had a case where only even numbers were considered, only odd numbers were considered, and all integers as a whole were considered. But I also know that you could have several subsequential limits, not just 1, 2, or 3. So I assume there's some sort of technique to construction a proper sequence, just not sure how I would. Do you think there's just a way to tack something on to the above function that might produce infinity?

Answer 6

You do need -1/2.

Maybe, and you'll love this: \(a_{n} = -\dfrac{3}{2} + \dfrac{5}{2}(-1)^{n} + \dfrac{1}{n\;mod\;2}\)

It blows up for every even number. Hmmm... Then we never get +2.

More thinking!

Answer 7

Oops, flipped which limit was negative and positive, you're right :P I like the mod idea. The only thing I worry about is if we're allowed to use it. We've kind of been restricted in what we can and cannot use. Often times if it hasn't been formally defined in the class then we can't use it. I suppose I could ask, usually the modulo idea definitely gives more options.

Since we're expanding into different types of operations and functions, you think, if we're allowed to use them, that sines and cosines are applicable for something like this?