Let $a_n\geq 0$ be a sequence of non-negative numbers. Consider the following two statements: $$ \text{(I)}\qquad\qquad \lim_{n\to\infty} \frac{1}{n^2}\sum_{i=1}^n a_i =0, $$ $$ \text{(II)}\qquad\qquad\qquad \sum_{n=1}^\infty \frac{a_n}{n^2}<\infty. $$
Questions: Does (I) imply (II)? Does (II) imply (I)? Otherwise, please provide counterexamples.
Motivation: Both statements occur in the context of the law of large numbers for non-identically distributed random variables. With $a_n=\mathrm{Var}(X_n)$, one can conclude the weak LLN if the $X_n$ are pairwise uncorrelated and condition (I) holds. The strong LLN can be concluded if the $X_n$ are stochastically independent and condition (II) holds. Therefore, one might expect that (II) implies (I).