Does there exist a sequence $a_n\in \mathbb{R}^{>0}$ such that $$\liminf_{N\rightarrow \infty}~~~ \frac{1}{N}\sum_{n=1}^Na_n >0$$
and $$\sum_{n=1}^\infty \frac{1}{a_n\cdot n^2} = \infty~~?$$
P.S. I posted the logical next question here Does there exist a non-increasing sequence with these properties?
Let $a_n= 1/n^2$ when $n$ is even, and $1$ when $n$ is odd. Then the average converges to $1/2$ and the sum is $\infty$.