What is a limit of a sequence?

[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

 

[arildno]

What is a limit of a sequence?

 

[teng125]

limit n to infinity

 

[ziad1985]

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.

 

[teng125]

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

 

[ziad1985]

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.

 

[arildno]

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

 

[Isma]

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

 

[arildno]

Not at all.

 

[Isma]

why is that?

 

[arildno]

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

 

[Isma]

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

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

 

[Isma]

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

 

[VietDao29]

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

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?

 

[arildno]

Besides, a sequence IS a function.

 

[benorin]

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

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

 

[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 [itex]a_{n}[/tex] if for any $\epsilon>0$ 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?
What about any other number?
Answer those questions, and you have the answer to your own question.