极限的证明题

一、设数列$\big\{a_{n}\big\}$ 恒满足不等式$\sqrt{n}|a_{n}|\leq 3，n=1,2,...$试证明

$$\lim_{n\to \infty} \frac{1}{n^{3} }\left [\left(\sum_1^n a_{i} \right) ^{2}+\left(\sum_2^n a_{i}  \right) ^{2}+\cdots+\left(\sum_n^n a_{i}  \right) ^{2}\right]=0$$

 

二、设正项数列$\big\{a_{n} \big\}$满足$\lim\limits_{n\to \infty}\frac{a_{n} }{a_{n-1} }=s>0$，试求

$$\lim_{n\rightarrow ∞}\left (\frac{\sqrt[n]{a_{1}a_{2}......a_{n} } }{a_{n} }\right )^{\frac{1}{n} }$$