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Math Help - Sequence

  1. #1
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    Sequence

    Say I have a sequence where for each positive integer n, a_{n+1}=a_n- \frac {1}{n}. Is there an a_1 I can choose which will make this a convergent sequence?

    It seems to me that if one choice of a_1 will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by spoon737 View Post
    Say I have a sequence where for each positive integer n, a_{n+1}=a_n- \frac {1}{n}. Is there an a_1 I can choose which will make this a convergent sequence?

    It seems to me that if one choice of a_1 will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.
    What have you gotten from this recurrence relationship? Have you used it to find anything yet?
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  3. #3
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    Quote Originally Posted by spoon737 View Post
    Say I have a sequence where for each positive integer n, a_{n+1}=a_n- \frac {1}{n}. Is there an a_1 I can choose which will make this a convergent sequence?

    It seems to me that if one choice of a_1 will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.
    Note that if a_1=a then a_n = a - \sum_{k=1}^n \frac{1}{k}. This leads to a harmonic series.
    Last edited by ThePerfectHacker; May 14th 2008 at 08:11 PM.
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  4. #4
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    Oh wow, I can't believe I missed that. So it can't converge at all then.
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