Vector Space of divergent sequences

**tintin2006** · January 30th 2011, 02:24 PM

Consider the set of divergent sequences:
$V=\{f|f:N \to R, \lim_{n \to \infty} f \mbox{ does not exists or is } \pm \infty \}$
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?

**dwsmith** · January 30th 2011, 02:27 PM

Originally Posted by tintin2006

Consider the set of divergent sequences:
$V=\{f|f:N \to R, \lim_{n \to \infty} f \mbox{ does not exists or is } \pm \infty \}$
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.

**Plato** · January 30th 2011, 02:35 PM

Originally Posted by tintin2006

Consider the set of divergent sequences:
$V=\{f|f:N \to R, \lim_{n \to \infty} f \mbox{ does not exists or is }\pm \infty \}$
Is this a vector space? Explain.

Is it true that if $s_n=(-1)^n~\&~t_n=(-1)^{n+1}$ then both are in $V~?$

What about their sum?

**tintin2006** · January 30th 2011, 02:50 PM

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

**tintin2006** · January 30th 2011, 03:05 PM

Originally Posted by Plato

Is it true that if $s_n=(-1)^n~\&~t_n=(-1)^{n+1}$ then both are in $V~?$

What about their sum?

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

**Plato** · January 30th 2011, 03:14 PM

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 $u_n=0$ .
That is not in V.

**HallsofIvy** · January 30th 2011, 03:18 PM

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, $\lim_{n\to\infty}s_n+ t_n$ is NOT divergent. Actually write out four or five terms of the sequence and look at the sum $s_n+ t_n$ .

**tintin2006** · January 30th 2011, 04:34 PM

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, $\lim_{n\to\infty}s_n+ t_n$ is NOT divergent. Actually write out four or five terms of the sequence and look at the sum $s_n+ t_n$ .

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

**dwsmith** · January 30th 2011, 04:42 PM

Originally Posted by tintin2006

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

Correct.

You could have done this too

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

Thread: Vector Space of divergent sequences

LinkBack

Thread Tools

Display

Vector Space of divergent sequences

Similar Math Help Forum Discussions

divergent sequences

Are the sum and/or product of two divergent sequences divergent?

Two divergent sequences whose product converges

divergent sequences

Convergent/Divergent Sequences...