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Prove that, if $a_k \geq 0$ for large $k$, and that $$\sum_{k=1}^\infty \frac{a_k}{k}$$ converges. Prove that $$\lim_{j \rightarrow \infty} \sum_{k=1}^\infty \frac{a_k}{j+k} = 0$$

I believe I cant write $$\sum_{k=1}^\infty \frac{a_k}{j+k} \leq \sum_{k=1}^\infty \frac{a_k}{k}.$$ However, taking limit does not give me the answer.

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marked as duplicate by Botond, Sil, Clement C., Community Aug 11 '18 at 16:34

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    $\begingroup$ The claim is false. Take $a_k = 1$ for all $k$: the series $\sum_{k\geq 1} \frac{1}{k}$ does not converge. $\endgroup$ – Clement C. Aug 11 '18 at 16:27
  • $\begingroup$ Some info is missing. $\sum_{k=1}^\infty \frac{1}{k}$ diverges $\endgroup$ – Adam Aug 11 '18 at 16:28
  • $\begingroup$ Clem you beat me to it. Seems he is still editing the question... $\endgroup$ – Adam Aug 11 '18 at 16:29
  • $\begingroup$ No, that is just bad edit of someone else... $\sum_{k=1}^\infty \frac{a_k}{k}$ is supposed to converge by assumption $\endgroup$ – Sil Aug 11 '18 at 16:29
  • $\begingroup$ @ClementC. I think the condition $ \exists \sum\limits_{k \in \mathbb{N}} \frac{a_k}{k}$ was also given, and he want to prove that $\lim\limits_{j \to \infty} \sum\limits_{k \in \mathbb{N}} \frac{a_k}{j+k}=0$. $\endgroup$ – Botond Aug 11 '18 at 16:30