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Complex infinite series

  1. Jan 27, 2006 #1
    i need a little help with this problem:

    determine if the infinite series converges or diverges.
    summation (from n=1..infinity) {1/(n^2+i^n)}

    I first applied the ration test to this series and got

    (n+1)^2 + i^(n+1) / [n^2 + i^n]

    i then multiplied top and bottom by (n^2 - i^n)
    which gave

    [{(n+1)^2 + i^(n+1) }* (n^2-i^n) ]/{n^4 - i^2n}

    this is where i get stuck, i cant seem to simplify it any further.
    if some one can give me some advice on it, it would be greatly appreciated.

  2. jcsd
  3. Jan 28, 2006 #2

    matt grime

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    the ratio test is the best way to do it. but when you've got an expression like


    all you need to do is divide every term by n^2.
  4. Jan 28, 2006 #3


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    Also don't forget that it is the limit of the absolute value of the ratio that counts.
  5. Jan 28, 2006 #4
    I don't think the ratio test works here. A simple comparison (of the terms' absolute values) seems to work better.
    Last edited: Jan 28, 2006
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