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Math Help - Help with this series please

  1. #1
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    Help with this series please

    Hi can someone help me figure this out:



    I know its convergent, but I'm trying to figure out how to get it into the form:

    a/1-r

    If you notice, when n = 1, the first term is always 1. Then it becomes 1+7^2/8^2 all the way till n.

    I figured that r should be 1/8 since each preceding term differs by a factor of 1/8. Can't figure out a though. Any ideas?
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  2. #2
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    Quote Originally Posted by Kuma View Post
    Hi can someone help me figure this out:



    I know its convergent, but I'm trying to figure out how to get it into the form:

    a/1-r

    If you notice, when n = 1, the first term is always 1. Then it becomes 1+7^2/8^2 all the way till n.

    I figured that r should be 1/8 since each preceding term differs by a factor of 1/8. Can't figure out a though. Any ideas?

    \displaystyle{\frac{1+7^n}{8^n}<2\left(\frac{7}{8}  \right)^n , and the rightmost series converges (why?), so the comparison test

    yields that also the original series converges.

    Tonio
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  3. #3
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    if i split the series into two parts.

    (1/8)^n + (7/8)^n

    We can see they both converge since r < 1 for both series.
    But how can i figure out what they converge to?
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  4. #4
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    Quote Originally Posted by Kuma View Post

    I know its convergent,
    Yes is certainly is.
    So are \displaystyle \sum\limits_{k = 1}^\infty  {\frac{1}<br />
{{8^k }}} \;\& \,\sum\limits_{k = 1}^\infty  {\frac{{7^k }}<br />
{{8^k }}}
    They are rearrangeable, being absolutely convergent.
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  5. #5
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    Quote Originally Posted by Kuma View Post
    if i split the series into two parts.

    (1/8)^n + (7/8)^n

    We can see they both converge since r < 1 for both series.
    But how can i figure out what they converge to?


    The sum of an infinite geometric series with first element a_1 and constant

    quotient q\,,\,|q|<1 , is \displaystyle{\frac{a_1}{1-q}}

    Tonio
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