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Find supremum

  1. Jul 24, 2011 #1
    1. The problem statement, all variables and given/known data

    Let S:={1-(-1)^n /n: n in N}
    Find supS.


    3. The attempt at a solution

    Is this the right way to write the solution?
    Thanks!

    First, I want to show that 2 is an upper bound.
    For any n in N, 1-(-1)^n /n is less or equal to 2. Thus, n=2 is an upper bound.

    Second, I want to show that 2 is the least upper bound.
    Assume there is an arbitrary upper bound v, such that v<2. Then, however, v< 1-(-1)^n /n for any n in N. Thus, v is not an upper bound. Hence, supS=2.
     
  2. jcsd
  3. Jul 25, 2011 #2

    micromass

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    Why is 1-(-1)^n/n always less or equal than 2?

    Why is v < 1-(-1)^n/n for any n in N?? (this is not true) You only need v<1-(-1)^n for one n in N. Which one?
     
  4. Jul 25, 2011 #3

    Gib Z

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    Pretty close, hopefully the mistake was just a typo:

    "Then, however, v< 1-(-1)^n /n for any n in N."

    Replace "any" with "some" and it's fine. Can you see why?

    EDIT: Too slow lol.
     
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