Investigate the convergence or divergence of the sequence a1=c,a2=c+a1,an+1=c+an,c>0. I proved it is increasing. But...
Investigate the convergence or divergence of the sequence
I proved it is increasing. But I could not prove it is bounded above or unbounded. I guess it is divergence sequence. How can I do this?
Answer & Explanation
Beginner2022-01-16Added 33 answers
Claim: is increasing and bounded above by . Also Proof: By induction on . . Assume , we show . But by the inductive step. So by induction . We show that by induction also. For If , we're done since . Otherwise, , and squaring both sides: is clearly true. Thus . Assume , you have which is true by inductive step. So . Thus the claim is verified. This follows that the limit exists and call it M. Then Since
Beginner2022-01-17Added 35 answers
First, you need to study the fixed points of the function
as only these can be the limit of the sequence
Solving , we find
The sequence starts at and is nonnegative, so the only possible candidate for the limit is
Now, we need to prove that if .
Thus, the sequence is bounded from above by y and is increasing. As y is the only possible limit, it converges to y.
Plainmath is a platform aimed to help users to understand how to solve math problems by providing accumulated knowledge on different topics and accessible examples. Plainmath.net is owned and operated by RADIOPLUS EXPERTS LTD.