What is the difference between these sum notations? $\sum_i \sum_j a_i b_j \overset{?}{=} \sum_{i,j} a_i b_j$

Question

I saw both written in a textbook (cannot remember where exactly), but I do not know what the difference should be between these:

$$\sum_i \sum_j a_i b_j \overset{?}{=} \sum_{i,j} a_i b_j$$

Answer 1

One explicitly has order defined on the sum $\sum_i \sum_j a_i b_j = \sum_i (\sum_j a_i b_j) =\sum_j a_1 b_j + \sum_j a_2 b_j +\sum_j a_3 b_j +\cdots $

or even if one insists that there is no order for indexes then

$\sum_i \sum_j a_i b_j = a_{19265} \sum_j b_j + a_{1} \sum_j b_j + a_{991} \sum_j b_j +\cdots $

where as $\sum_{i,j} a_i b_j$ does not explicitly define any particular order e.g. $\sum_{i,j} a_i b_j = a_{189}b_0 + a_1b_{7676} + a_1b_1$ etc.

For convergence with some multidimensional series the order matters, where as with changing the order could converge to any value.