Hi,
I have the following questions about sequences:
1) Suppose a(n) > 0 for every n and the sum of all a(n) diverges.
Let s(n) = a(1)+....+a(n). Prove that b(n)=a(n) / s(n) diverges and does so slower than a(n).
2) Suppose a(n) > 0 for every n and the sum of all a(n) converges.
Let r(n) = a(n) + a(n+1) + ..... a(infinity). Prove that b(n) = a(n) / sqrt(r(n)) converges and does so slower than a(n).
What I did:
For the second question, I tried to give a proof by induction bu I think I failed.
First I tried to show that {b(n)} is strictly decreasing, then I tried to prove that {b(n)} is bounded below by induction. However, I was not able to show why it converges slower than a(n).
Additionally, I have no idea how to show that one of the sequences diverges and the other one converges.
I didn't solve the first question, as I know that there is something wrong with my answer to the second question.
Could you guys give me your thoughts on this?