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Math Help - Proof of convergence

  1. #1
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    Proof of convergence

    I need to prove that the sequence { a_n} converges if and only if \sum _{n=1}^{\infty} (a_{n+1} - a_n) converges. I think I've managed to prove that if the series converges then the sequence converges, but I'm having trouble with the other direction. I have a feeling I'm missing something really obvious, but any help would be appreciated.
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  2. #2
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    Quote Originally Posted by spoon737 View Post
    I need to prove that the sequence { a_n} converges if and only if \sum _{n=1}^{\infty} (a_{n+1} - a_n) converges. I think I've managed to prove that if the series converges then the sequence converges, but I'm having trouble with the other direction. I have a feeling I'm missing something really obvious, but any help would be appreciated.
    If \{ a_n \} converges it means |a_n - a_m| < \epsilon by using the Cauchy sequence. But then it means \left| \sum_{k=m}^{n-1} a_{k+1} - a_k\right| = |a_n - a_m| < \epsilon. Thus, the series satisfies the Cauchy condition for convergence.
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  3. #3
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    Thanks for the help, but we haven't learned the Cauchy convergence criterion for series yet, so I don't think I can use it. Is there another way I can approach this?
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  4. #4
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    Quote Originally Posted by spoon737 View Post
    Thanks for the help, but we haven't learned the Cauchy convergence criterion for series yet, so I don't think I can use it. Is there another way I can approach this?
    Well it is the same thing as for sequences. If s_n is the sequence of partial sums then |s_n - s_m| is that finite sum I wrote up there. All it is, is the ordinary Cauchy sequence but applied to a sequence of partial sums.
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  5. #5
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    Okay, that makes sense. Thanks again.
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