Homework Help: What is a limit of a sequence?

1. Feb 16, 2006

teng125

for lim n to infinity {2 [(-1)^n] },what should i write either converges or diverges because it tends to be -2 or +2

2. Feb 16, 2006

arildno

What is a limit of a sequence?

3. Feb 16, 2006

teng125

limit n to infinity

4. Feb 16, 2006

If i'm not mistaken, first thing you need to check in a series if the limit is unique , as in when the series limit is one number.
then you should check if that limit is zero or not , then you can apply usual methodes know to find out.
So in your case it diverge.

5. Feb 16, 2006

teng125

is it if there is (-1^n) and n limit to infinity means always diverges??

6. Feb 16, 2006

doesn't have to.
[(-1)^n]/n have a limit when n tends to be infinite 0 , this series converge.
I sense you don't know much about limits , maybe you didn't studied well that chapter.
it's fairly easy to know them.

7. Feb 16, 2006

arildno

Again, teng:
What is the definition of a limit?

8. Feb 17, 2006

Isma

"limit" is how a function's behavior changes when its argument(or variable) gets close to a certain value(or a point)

9. Feb 17, 2006

arildno

Not at all.

10. Feb 17, 2006

Isma

why is that?

11. Feb 17, 2006

arildno

Because what you wrote is meaningless.
Go back to your textbook and look up the definition of a limit to a sequence.

12. Feb 17, 2006

Isma

but ur question simply says
arildno :
What is the definition of a limit?

u dint specify it was limit to a sequence...

13. Feb 17, 2006

Isma

i just thought u were saying abt limit of a function....so sorry

14. Feb 17, 2006

VietDao29

Nah, this is neither the definition for limit of a function nor limit of a sequence...
You may want to look it up in your book. :)
By the way, are you teng125?

15. Feb 17, 2006

arildno

Besides, a sequence IS a function.

16. Feb 17, 2006

benorin

OK, the lim inf is -2 and the lim sup is +2, and thus the lim is...

17. Feb 18, 2006

arildno

teng:
Forget subsequences, lim infs and all that.
Those concepts won't help you a bit, because you betray an uncertainness abut the very concept of a limit in the first place.

Let us take a typical textbook definition:
"We say that a number L is a LIMIT of a sequence $a_{n}[/tex] if for any [itex]\epsilon>0$ there exists a number N, so that for any n>N, $|a_{n}-L|<\epsilon$"
Furthermore, a sequence is said to diverge if no such number L exists.

1. The first thing to note is that L (if it exists) is a NUMBER, and only that.
It is not a hand-wavy action by which we describe the function's behaviour. It is a number. Period.

2. Secondly, the definition should be regarded as a recipe of detemining whether or not an arbitrarily chosen L is a limit to the sequence or not.

3. Let us for convenience sake pick L=2 first, and check whether it can be said to be a limit to the sequence:

Consider the absolute valued difference: $$|2(-1)^{n}-2|$$
Now, when n is even, this difference equals 0, but when n is odd, the difference equals 4.
Thus, by picking $\epsilon<4$, and remembering that odd numbers can be arbitrarily big, we see that there cannot exist a number N so that for ANY n>N, the difference is less than $\epsilon$

Thus, 2 cannot be regarded as a limit L to our sequence.

Do you think -2 can be a limit?