Homework Help: Bounded sequences

1. Jan 15, 2008

sara_87

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. Jan 15, 2008

Mathdope

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
3. Jan 15, 2008

sara_87

oh okay
so for 1) the sequence will never go below 0 for all n so it is bounded right?

4. Jan 15, 2008

sara_87

oh sorry
M>0?
then it is not bounded

5. Jan 15, 2008

Mathdope

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