Does $\ \displaystyle \sum_{n\geq 0} a_n\ \text{ converges}\ \implies\ \lim_{n\to\infty}\left( a_0 a_n + a_1 a_{n-1} + a_2 a_{n-2} + a_{ \big\lfloor \frac{n}{2} \big\rfloor } a_{ \big\lceil \frac{n}{2} \big\rceil } \right)\ $ converges?
And if it doesn't converge for conditionally convergent series $\ \displaystyle \sum_{n\geq 0} a_n\ $, then does it converge for absolutely convergent series $\ \displaystyle \sum_{n\geq 0} a_n\ $?
If we try to use Cauchy-Schwarz, we get:
$$ \left( a_0 a_n + \ldots + a_{ \big\lfloor \frac{n}{2} \big\rfloor } a_{ \big\lceil \frac{n}{2} \big\rceil } \right)^2$$
$$ \leq \left( {a_0}^2 + {a_1}^2 + \ldots + {\left( a_{ \big\lfloor \frac{n}{2} \big\rfloor }\right)}^2 \right) \left( {a_n}^2 + {a_{n-1}}^2 + \ldots + {\left( a_{ \big\lceil \frac{n}{2} \big\rceil }\right)}^2 \right).$$
On the right-hand side, the first bracket could $\ \to \infty,\ $ and I'm not sure what you can say about the second bracket, but even if you could show that the second bracket $\ \to 0\ $ as $\ n\to\infty\ $ (which I am not sure of), the right hand side is still indeterminate, so C-S doesn't seem to help.
Also for example if $\ a_n = \frac{1}{n^2},\ $ then I'm not sure what $\ \lim_{n\to\infty}\left( a_0 a_n + a_1 a_{n-1} + a_2 a_{n-2} + a_{ \big\lfloor \frac{n}{2} \big\rfloor } a_{ \big\lceil \frac{n}{2} \big\rceil } \right)\ $ is. Perhaps this can be made into an integral? Although I'm not sure how to do this either.