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Infinite Series

  1. Oct 26, 2013 #1
    1. The problem statement, all variables and given/known data

    Determine whether the series diverges or converges.

    (1+2) / (1+3)+ ((1+2+4)/(1+3+9))+ ((1+2+4+8)/(1+3+9+27)) + ...


    3. The attempt at a solution

    I have split up the series into two (denominator and numerator):

    an = (1+2) + (1+2+4) + (1+2+4+8)+... = (1)n + 2n + 4(n-1) + ...
    bn = (1+3) + (1+3+9)+... = (1)n + (3)n + (9)(n-1)+... = (1)n + 3n + 9(n-1) + ...


    I don't know how to keep going. I suspect that the ratio test will come in handy later but am not sure how to apply it with the given series above. Any help would be appreciated. Thanks.
     
  2. jcsd
  3. Oct 26, 2013 #2

    Office_Shredder

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    How is evaluating an and bn supposed to help? an/bn is not the same as the partial sum of the series you have been given.

    You might want to think about a short way to write down
    1+2+4+8+16+...
    for any finite term.
     
  4. Oct 27, 2013 #3
    Well we can write 1+2+4+8+... as

    the sum from n=0 to n=inf of 2^n.

    And similarly we can write 1+3+9+...

    as the sum from n=0 to n=inf of 3^n.

    So can we say that the series is (2/3)^n? From n=0 to n=inf?
     
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