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

  • 1
    $\begingroup$ Have you tried induction? On the surface, this looks like an induction problem. $\endgroup$ – Clayton Nov 8 '18 at 19:22
  • $\begingroup$ It seems I couldn't fit this question in full to the title due to character limits. $\endgroup$ – davidh Nov 8 '18 at 19:25
  • $\begingroup$ oh, I misunderstood... will edit. $\endgroup$ – davidh Nov 8 '18 at 19:26

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