$\sum \limits_{n=1}^{\infty}{a_n^2}$ converges $\implies \sum \limits_{n=1}^{\infty}{\frac{a_n}{n}}$ [duplicate]

Solution 1:

Apply Hölder's Inequality (Cauchy-Schwarz if you are not familiar with Hölder) $$ \sum_{n=1}^\infty \left|a_n\cdot \frac1n \right | \leq \left(\sum_{n=1}^\infty a_n^2\right)^{1/2} \left(\sum_{n=1}^\infty\frac1{n^2} \right)^{1/2} < \infty $$ so the series is absolutely convergent.

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