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Homework Help: Problem with convergent sequences

  1. Jul 3, 2012 #1
    Hi, I have the following problem and have done the first two questions, but I don't know how to solve the last two. Thanks for any help you can give me!

    1. The problem statement, all variables and given/known data
    Let [itex]a_{n}\rightarrow a[/itex], [itex]b_{n}\rightarrow b[/itex] be convergent sequences in [itex]\Re[/itex]. Prove, or give a counterexample to, the following statements:

    A) [itex]a_{n}[/itex] is a monotone sequence;
    B) if [itex]a_{n}>b_{n}+1/(n^3+4)[/itex], then [itex]a>b[/itex];
    C) if [itex]a_{n}>((n^3+1)/(2n^3+1))b_{n}[/itex], then a>b;
    D) if [itex]s_{n}=(1/n)(a_1+...+a_n)[/itex], then [itex]s_n \rightarrow a[/itex].

    2. Relevant equations

    3. The attempt at a solution

    I have solved the first two. For A I have given the counterexample [itex]a_n=sin(n)/n[/itex] and for B I have used the fact that as n goes to infinity, [itex]1/(n^3+4)[/itex] approaches 0, which would give [itex]a_n > b_n[/itex], which is a>b when n goes to infinity.

    I have tried the same thing with C, but it gives me [itex]a>(1/2)b[/itex], which doesn't lead me anywhere, I think. And for D, I think that as n goes to infinity, [itex]s_n[/itex] will be close to [itex]a_n[/itex] because [itex]s_n ≈ (1/n)*n*a_n[/itex], which is the same as saying [itex]s_n \rightarrow a[/itex]. However, I don't know if this is correct, and if it is, how am I supposed to express it?

    Thanks a lot!
  2. jcsd
  3. Jul 3, 2012 #2


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    Think about that last statement if ##a_n = \frac 2 {n^3+4}## and ##b_n=0##.
  4. Jul 4, 2012 #3

    So that would give me [itex]2/(n^3+4)>1/(n^3+4)[/itex], which holds when n goes to infinity.

    So could I use something similar for part C then, something like [itex]b_n=0[/itex] or [itex]b_n=1[/itex]? With [itex]b_n=1[/itex], I could have [itex]a_n=(2n^3+1)/(2n^3+1)[/itex]?
  5. Jul 4, 2012 #4
    However, they would not hold for negative values of n? I am confused :/
  6. Jul 4, 2012 #5


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    I'm not sure you understand my example and I don't know what you mean when you say it "holds when n goes to infinity". The ##a_n## and ##b_n## in my example go ##a=0## and ##b=0## respectively. The ##a_n>b_n## does not hold in the limit.
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