Prove limit of a sum is $0$ [duplicate]

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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.

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

• The claim is false. Take $a_k = 1$ for all $k$: the series $\sum_{k\geq 1} \frac{1}{k}$ does not converge. – Clement C. Aug 11 '18 at 16:27
• Some info is missing. $\sum_{k=1}^\infty \frac{1}{k}$ diverges – Adam Aug 11 '18 at 16:28
• No, that is just bad edit of someone else... $\sum_{k=1}^\infty \frac{a_k}{k}$ is supposed to converge by assumption – Sil Aug 11 '18 at 16:29
• @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 $\lim\limits_{j \to \infty} \sum\limits_{k \in \mathbb{N}} \frac{a_k}{j+k}=0$. – Botond Aug 11 '18 at 16:30