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Bounded sequences

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data
    how do show whether the following sequences are bounded?
    1) {an}=sqrt(n)/1000
    2) {an}=(-2n^2)/(4n^2 -1)
    3) {an}=n/(2^n)
    4) {an}=(ncos(npi))/2^n

    2. Relevant equations
    i have to show whether the sequences are bounded by a number but i dont know how to find that number. for part (4) i have to use the sandwich theorem.

    3. The attempt at a solution
    1) it's not bounded since it will continue to increase to infinity.
    but i dont know how to do the rest. can someone help please?
    thank you very much
  2. jcsd
  3. Jan 15, 2008 #2
    If a sequence (a_n) is bounded then there exists an M > 0 such that |a_n| (< or =) M. On the other hand if a sequence is unbounded then for all M > 0 there exists an N such that if n > N, |a_n|> M.

    Now, if I give you a number M can you show that each sequence will either (a) never exceed or (2) eventually exceed M? Think of a specific example first such as M = 100. Can, for instance, the first sequence ever exceed 100? What value of N would guarantee it?
    Last edited: Jan 15, 2008
  4. Jan 15, 2008 #3
    oh okay
    so for 1) the sequence will never go below 0 for all n so it is bounded right?
  5. Jan 15, 2008 #4
    oh sorry
    then it is not bounded
  6. Jan 15, 2008 #5
    For (2) and (3) replace n with x and think of them as functions. Do they achieve maximums/minimums? Do they have limits as x (i.e. n) -> infinity? For (4) use the fact that |cos(n pi)| = 1 for all integers n, and do the same.
    Last edited: Jan 15, 2008
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