Solving Bounded Sequences Homework - How to Find Bounding Number

[sara_87]

Homework Statement

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

Homework Equations

i have to show whether the sequences are bounded by a number but i don't know how to find that number. for part (4) i have to use the sandwich theorem.

The Attempt at a Solution

1) it's not bounded since it will continue to increase to infinity.
but i don't know how to do the rest. can someone help please?
thank you very much

 

[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?

 

[sara_87]

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

 

[sara_87]

oh sorry
M>0?
then it is not bounded

 

[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.