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Test for convergence of the series

  1. Feb 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Q)
    Summation from 1 to infinity
    (1+(-1)^i) / (8i+2^i)

    This series apparently converges and I can't figure out why.


    2. Relevant equations



    3. The attempt at a solution

    (1+(-1)^i) / i(8+2^i/i)

    Taking the absolute value of the above generalization:

    2/i(8+2^i/i)

    Rearranging that would give:

    2/i * (1/(8+2^i/i)

    Now I thought that since 2/i would diverge, the entire series should diverge.

    Comparing it to the geometric series (1/2^i) implies converges, but I don't know what's wrong with the above
     
  2. jcsd
  3. Feb 26, 2012 #2

    vela

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    I don't understand your reasoning here. It's like saying
    $$\sum_{n=1}^\infty \frac{1}{2^n}$$ diverges because you can write it as
    $$\sum_{n=1}^\infty \left(\frac{1}{2}\times\frac{1}{2^{n-1}}\right)$$ and $$\sum_{n=1}^\infty \frac{1}{2}$$ diverges.
     
  4. Feb 26, 2012 #3
    Point. I just checked my textbook and it appears I'd mistakenly thought that just because Summation 1-infinity (ai+bi) equals summation 1-infinity (ai) + Summation 1-infinity (bi), I thought the same would be true for multiplication.

    Thanks for clearing it up.
     
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