Let $a_n$ be a sequence of non-negative real numbers, such that $\sum a_n$ diverges. For $n\ge1$, let $s_n = a_1 + ... + a_n$.

Prove that for all $N\ge1$ and all $n\ge1$, $\sum_{k=1}^{n}\frac{a_{N+k}}{s_{N+k}} \ge 1 - \frac{s_N}{s_{N+n}}$

I tried to manipulate the terms to apply comparison test, but I couldn't really get anywhere with this problem. Thanks

New contributor
davidh is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
  • 1
    Have you tried induction? On the surface, this looks like an induction problem. – Clayton Nov 8 at 19:22
  • It seems I couldn't fit this question in full to the title due to character limits. – davidh Nov 8 at 19:25
  • oh, I misunderstood... will edit. – davidh Nov 8 at 19:26

Your Answer

davidh is a new contributor. Be nice, and check out our Code of Conduct.
 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.