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Math Help - Vector Space of divergent sequences

  1. #1
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    Vector Space of divergent sequences

    Consider the set of divergent sequences:
    V=\{f|f:N \to R, \lim_{n \to \infty} f <br />
\mbox{ does not exists or is }<br />
\pm \infty<br />
 \}
    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?
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    Quote Originally Posted by tintin2006 View Post
    Consider the set of divergent sequences:
    V=\{f|f:N \to R, \lim_{n \to \infty} f <br />
\mbox{ does not exists or is }<br />
\pm \infty<br />
 \}
    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.
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  3. #3
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    Quote Originally Posted by tintin2006 View Post
    Consider the set of divergent sequences:
    V=\{f|f:N \to R, \lim_{n \to \infty} f <br />
\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?
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    Okay, say I'll use the Additive Closure: u+v \in V one:
    <br />
\lim_{n \to \infty }f_{1} + \lim_{n \to \infty }f_{2} = \infty <br />
which is not in R? Or did I misunderstand something?
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    Quote Originally Posted by Plato View Post
    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.
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  6. #6
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    Quote Originally Posted by tintin2006 View Post
    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.
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  7. #7
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    Quote Originally Posted by tintin2006 View Post
    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.
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    Quote Originally Posted by HallsofIvy View Post
    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.
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  9. #9
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    Quote Originally Posted by tintin2006 View Post
    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
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