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If $\{u_n\}$ is a sequence of real numbers and the series $\displaystyle\sum_{n=1}^{\infty}u_n^2$ is convergent, prove that the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is absolutely convergent.

I have proved the series $\displaystyle\sum_{n=1}^{\infty}\frac{u_n}{n}$ is convergent, but how to prove it is absolutely convergent?

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marked as duplicate by Martin R, Gabriel Romon, Math1000, Community May 16 '18 at 10:05

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One may use$$ 2\cdot \frac{|u_n|}n \le |u_n|^2+\frac{1}{n^2},\quad n\ge1. $$

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  • $\begingroup$ (proof?)....... $\endgroup$ – James May 16 '18 at 8:56
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    $\begingroup$ @JimmySabater We have$$ 0\le\left( |u_n|-\frac1n\right)^2$$ $\endgroup$ – Olivier Oloa May 16 '18 at 8:57
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Apply Cauchy-Schwarz:$$\sum_{i=1}^N\left|\frac{u_n}n\right|\leqslant\sqrt{\sum_{i=1}^n{u_i}^2}\sqrt{\sum_{i=1}^n\frac1{i^2}}.$$

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