# Convergence of &#x2211;<!-- ∑ --> a n </msub> <msqrt> n </m

Convergence of $\sum \frac{{a}_{n}}{\sqrt{n}}$ given that $\sum {a}_{n}^{2}$ converges
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Dayana Zuniga
Let ${a}_{n}=\frac{1}{\sqrt{n}\mathrm{log}n}$. Then $\sum _{n=2}^{\mathrm{\infty }}{a}_{n}^{2}=\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n{\mathrm{log}}^{2}n}$ which converges by Cauchy's condensation test, but
$\sum _{n=2}^{\mathrm{\infty }}\frac{{a}_{n}}{\sqrt{n}}=\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\mathrm{log}n}$
and this diverges, again by Cauchy's condensation test.