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Thread: Limit of a simple sequence

  1. #1
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    Limit of a simple sequence

    Prove:

    The sequence An = 1/(n + 1) + 1/(n + 2) +...+ 1/(2n) has a limit.
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  2. #2
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    $\displaystyle n + 1 < n + 2 < \cdots < 2n \Leftrightarrow \frac{1}{2n} < \frac{1}{2n - 1} < \cdots < \frac{1}{n+1} $.

    Notice that your sum has n amount of terms, and so $\displaystyle 0 < \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} < \frac{n}{2n} = \frac{1}{2} $.

    Now show that it's monotonic, and you're finished.
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  3. #3
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    Thanks allot for your help.

    Quick question:

    Would it not be <=?
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  4. #4
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    you're working for $\displaystyle n\ge1,$ so everything is positive.
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  5. #5
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    Really I was referring to the <= n/2n = 1/2. This still holds though?

    Thanks
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