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Thread: Geometric series

  1. March 14th 2006, 05:39 PM #1
    Stuart
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    Geometric series

    How would I set up a equation to find the sum of...


    4
    E log24^n
    n=1
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  2. March 14th 2006, 08:50 PM #2
    CaptainBlack
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    Quote Originally Posted by Stuart
    How would I set up a equation to find the sum of...


    4
    E log24^n
    n=1
    It all depends on what you are actualy asking.

    Interpretation 1:

    <br /> \sum_{n=1}^4 \log(24^n)=\log(24)+\log(24^2)+\log(24^3)+\log(24^ 4) =\log(24^{10})=10\log(24)<br />

    Interpretation 2:

    <br /> \sum_{n=1}^4 \log_2(4^n)=\sum_{n=1}^4 \log_2(2^{2n})=\sum_{n=1}^4 2n\log_2(2)=\sum_{n=1}^4 2n=20<br />

    RonL
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  3. March 15th 2006, 09:30 AM #3
    ThePerfectHacker
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    Interpretation 3:
    \sum^4_{n=1}\log^n(24)
    but this one is rather nasty.
    Last edited by ThePerfectHacker; March 15th 2006 at 09:33 AM.
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  4. March 15th 2006, 12:15 PM #4
    CaptainBlack
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    Quote Originally Posted by ThePerfectHacker
    Interpretation 3:
    \sum^4_{n=1}\log^n(24)
    but this one is rather nasty.

    <br /> \log(24)+\log(\log(24))+ \log(\log(\log(24)))+\log(\log(\log(\log(24))))<br />

    which my calculator will do.

    RonL
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  5. March 15th 2006, 01:41 PM #5
    ThePerfectHacker
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    Quote Originally Posted by CaptainBlack
    <br /> \log(24)+\log(\log(24))+ \log(\log(\log(24)))+\log(\log(\log(\log(24))))<br />

    which my calculator will do.

    RonL
    WHAT ?!?!
    It means,
    \log^n x=(\log x)^n
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  6. March 15th 2006, 08:40 PM #6
    CaptainBlack
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    Quote Originally Posted by ThePerfectHacker
    WHAT ?!?!
    It means,
    \log^n x=(\log x)^n
    I'm afraid it does not. There is a common abuse of notation with
    trig functions which might lead you to think this, but that is the
    aberrant usage not this:

    <br /> f^2 (x)=f(f(x)),\ f^3(x)=f(f(f(x))),\ \dots<br />

    The abuse of notation that leaves:

    <br /> f^n(x)=(f(x))^n<br />

    is in general to be deprecated as it leaves us with no compact
    notation for the iterates of functions

    RonL
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