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Math Help - some questions on convergence of sequences

  1. #1
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    some questions on convergence of sequences

    Let (a_n) be a sequence.

    (i) Prove that if \sum_{n=1}^{\infty}{a_n} converges, then \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})} also converges.

    (ii) Prove that if \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})} converges and a_n \rightarrow {0}, then \sum_{n=1}^{\infty}{a_n} converges.


    For (i), can i use the concept of subsequences to tackle this question.
    For (ii), how do I start this question?

    Thanks in advance!
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  2. #2
    Grand Panjandrum
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    Re: some questions on convergence of sequences

    Quote Originally Posted by alphabeta89 View Post
    Let (a_n) be a sequence.

    (i) Prove that if \sum_{n=1}^{\infty}{a_n} converges, then \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})} also converges.
    I'm not sure this is true. What happens if a_n=(-1)^{n-1}/n, then \sum_{n=1}^{\infty}{a_n} converges but I don't think \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})} does.

    CB
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  3. #3
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    Re: some questions on convergence of sequences

    Quote Originally Posted by CaptainBlack View Post
    I'm not sure this is true. What happens if a_n=(-1)^{n-1}/n, then \sum_{n=1}^{\infty}{a_n} converges but I don't think \sum_{n=1}^{\infty}{(a_{2n-1}+a_{n})} does.

    CB
    hmm? I think it still works for this sequence.
    Last edited by alphabeta89; January 11th 2012 at 05:32 PM.
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  4. #4
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    Re: some questions on convergence of sequences

    Anyone has any idea on starting this question?
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