real analysis – Question of the convergence of an infinite number of series with special conditions

I'm trying to think about the following types of problems and, in fact, if we could get results, they could be applied inside mathematics, even to unresolved problems.

I will not write any potential applications now, but will pass to the direct presentation of a problem.

I'm studying (though, no conclusions yet) the sum $$ sum_ {3 leq a_ {1,1}, a_ {1,2} <+ infty} frac {1} {a_ {1,1} a_ {1,2}} + sum_ {3 a_ {2,1}, a_ {2,2}, a_ {2,3} <+ infty} frac {1} {a_ {2,1} a_ {2,2} a_ {2,3 }} + … + sum_ {3 leq a_ {n, 1}, a_ {n, 2}, …, a_ {n, n + 1} <+ infty} frac {1} { a_ {n, 1} a_ {n, 2}, … a_ {n, n + 1}} + … $$.

or $ {a_ {1,1} a_ {1,2} } $ and $ {a_ {2,1} a_ {2,2} a_ {2,3} } $ and and $ {a_ {n, 1} a_ {n, 2}, … a_ {n, n + 1} } $ and … are all zero density $ mathbb N $ and every term in every sequence is strange.

Does all this imply the convergence of this infinite sum of infinite sums?