Suggestion of how to do it, please.

Suppose $\{a_n\}$ is a succession in $\mathbb R ^+$ such that $\sum a_n$ diverges, and if $s_n = \sum\limits_{k=1}^n{a_k}$. show that $\frac {a_{N+1}}{s_{N+1}} +...+\frac {a_{N+k}}{s_{N+k}}\geq 1 -\frac {s_{N}}{s_{N+k}} $

and infer that $\sum\limits_{n=1}^\infty{\frac{a_n}{s_n}}$ diverge.



marked as duplicate by Martin R, Namaste, Christopher, hardmath, Nosrati Oct 12 '18 at 18:27

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$$\frac {a_{N+1}}{s_{N+1}} +...+\frac {a_{N+k}}{s_{N+k}}\geq\frac {a_{N+1}}{s_{N+k}} +...+\frac {a_{N+k}}{s_{N+k}}=$$ $$=\frac{s_{N+k}-s_N}{s_{N+k}}= 1 -\frac {s_{N}}{s_{N+k}} $$

  • $\begingroup$ is $\frac {a_{N+1}}{s_{N+1}} +...+\frac {a_{N+k}}{s_{N+k}}\geq 1 -\frac {s_{N}}{s_{N+k}} $ $\endgroup$ – VERA Oct 12 '18 at 6:04
  • $\begingroup$ @VERA Yes, of course. See my solution. $\endgroup$ – Michael Rozenberg Oct 12 '18 at 6:05

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