- #1
gilabert1985
- 7
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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!
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].
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!
Homework Statement
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].
Homework Equations
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!