正项级数的一些小问题（1）

Let $\sum_{n=1}^{\infty}a_{n}$ and $\sum_{n=1}^{\infty}b_{n}$ ,both of them are convergent series with positive terms .For every $\mu+\nu\geq1,\mu>0,\nu>0:$ $$\text{The}\quad (1)\sum_{n=1}^{\infty}\frac{a_{n}^{\mu}}{n^{\nu}}, \quad(2)\sum_{n=1}^{\infty}a_{n}^{\mu}b_{n}^{\nu}\quad\text{are convergence?} $$ $\textbf{A}.\quad$ For every $\mu+\nu>1,\mu>0,\nu>0:$ $$\text{The}\quad (1)\sum_{n=1}^{\infty}\frac{a_{n}^{\mu}}{n^{\nu}}, \quad(2)\sum_{n=1}^{\infty}a_{n}^{\mu}b_{n}^{\nu}\quad\text{are convergence.(Hint:Young's Inequality)}$$ $\textbf{B}.\quad$ For every $\mu+\nu=1,\mu>0,\nu>0:$ $$(1)\text{Let} \mu=\nu=\frac{1}{2},\mu_{n}=\frac{1}{n\log^{2}n},\text{ then} \quad\sum_{n=1}^{\infty}\frac{a_{n}^{\mu}}{n^{\nu}} \quad\text{is divergence}.$$ $$(2)\text{Using H$\ddot o$lder's Inequality ,Let$p=\frac{1}{\mu},q=\frac{1}{\nu}$,$\frac{1}{p}+\frac{1}{q}=1$} ,\text{ then} \quad\sum_{n=1}^{\infty}a_{n}^{\mu}b_{n}^{\nu} \quad \text{is convergence}.\diamondsuit$$ $\text{SXFXJC vol.2 CSB p.148}$