the largest sum is always going to be: add up all positive integers in the array and exclude negatives. as adding is communicative, order of the array and sequence of the add operation doesn't make any difference.

While this certainly gives the largest sum of a sub(multi)set of the entries of the array, it needn't satisfy the conditions of the problem, since that required the summands to be consecutive terms of the array. (Incidentally, addition is commutative.)