Thread: Vector Space of divergent sequences

1. Vector Space of divergent sequences

Consider the set of divergent sequences:
$\displaystyle V=\{f|f:N \to R, \lim_{n \to \infty} f \mbox{ does not exists or is } \pm \infty \}$
Is this a vector space? Explain.
I think this is not a vector space because if the limit of the sequence is DNE, then it is not in R. Is that correct?

2. Originally Posted by tintin2006
Consider the set of divergent sequences:
$\displaystyle V=\{f|f:N \to R, \lim_{n \to \infty} f \mbox{ does not exists or is } \pm \infty \}$
Is this a vector space? Explain.
I think this is not a vector space because if the limit of the sequence is DNE, then it is not in R. Is that correct?
You have to prove all the axioms or find one that doesn't work.

3. Originally Posted by tintin2006
Consider the set of divergent sequences:
$\displaystyle V=\{f|f:N \to R, \lim_{n \to \infty} f \mbox{ does not exists or is }\pm \infty \}$
Is this a vector space? Explain.
Is it true that if $\displaystyle s_n=(-1)^n~\&~t_n=(-1)^{n+1}$ then both are in $\displaystyle V~?$

4. Okay, say I'll use the Additive Closure: $\displaystyle u+v \in V$ one:
$\displaystyle \lim_{n \to \infty }f_{1} + \lim_{n \to \infty }f_{2} = \infty$ which is not in R? Or did I misunderstand something?

5. Originally Posted by Plato
Is it true that if $\displaystyle s_n=(-1)^n~\&~t_n=(-1)^{n+1}$ then both are in $\displaystyle V~?$

It seems that both are in V (as in divergent sequences) and their sum is in V too.

6. Originally Posted by tintin2006
It seems that both are in V (as in divergent sequences) and their sum is in V too.
The sum is the sequence $\displaystyle u_n=0$.
That is not in V.

7. Originally Posted by tintin2006
It seems that both are in V (as in divergent sequences) and their sum is in V too.
Except that that last statement is untrue.

If it were true that those are divergent sequences whose sum was also a divergent sequence, that would prove nothing at all. In particular, it would not prove that "the sum of any/ two divergent sequences. Plato suggested you look at those two sequences because they give a counter-example. In particular, $\displaystyle \lim_{n\to\infty}s_n+ t_n$ is NOT divergent. Actually write out four or five terms of the sequence and look at the sum $\displaystyle s_n+ t_n$.

8. Originally Posted by HallsofIvy
Except that that last statement is untrue.

If it were true that those are divergent sequences whose sum was also a divergent sequence, that would prove nothing at all. In particular, it would not prove that "the sum of any/ two divergent sequences. Plato suggested you look at those two sequences because they give a counter-example. In particular, $\displaystyle \lim_{n\to\infty}s_n+ t_n$ is NOT divergent. Actually write out four or five terms of the sequence and look at the sum $\displaystyle s_n+ t_n$.
Ah! I get it, sorry how thought-less of me. $\displaystyle \sum_{i=0}^n (-1)^{i}+ (-1)^{i+1} = [1 + (-1)] + [-1 + 1] + ... = 0$. The limit of their sum is convergent which is not in V. Thank you.

9. Originally Posted by tintin2006
Ah! I get it, sorry how thought-less of me. $\displaystyle \sum_{i=0}^n (-1)^{i}+ (-1)^{i+1} = [1 + (-1)] + [-1 + 1] + ... = 0$. The limit of their sum is convergent which is not in V. Thank you.
Correct.

You could have done this too

$\displaystyle s_n+t_n=(-1)^n(1-1)=0$