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Alternating series test for convergence

  1. Oct 26, 2015 #1
    1. The problem statement, all variables and given/known data

    20151026_230927.jpg

    2. Relevant equations

    20151026_230940.jpg

    3. The attempt at a solution

    I don't get how they got what's stated in the above picture. Where does 1/2 and n/(n + 1) come from? Can't you just show that an + 1 ≤ an?
     
  2. jcsd
  3. Oct 26, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    That's what they do. The logic is easier to follow if you start with the result and reduce this to something true (look at the steps in reverse order), but this direction works as well. The first line is then a clever guess what will be needed later (and the statement is clearly true).
     
  4. Oct 26, 2015 #3
    A lot easier to follow that direction, thanks! The final step has 2n and 2n - 1 instead of (-2)n and (-2)n - 1 which I don't get.
     
  5. Oct 26, 2015 #4

    mfb

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    Staff: Mentor

    The test compares the magnitude of the sequence elements only, powers of -1 don't change the magnitude.
    The sign has to be checked separately (easy here).
     
  6. Oct 26, 2015 #5
    Oh okay! Thank you :smile:
     
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