I think that is the key point (I wish it was nearer the top!). If i interpret you correctly, all you are really doing here is changing the order of summation, and you know that will not affect the value of the sum. (i think that is also the gist of what you were saying)Both functions map the same set of natural numbers to the same subset of reals.
To state it rigourously, you could show that there is a 1-1 mapping between the terms in the first sum and the terms in the second sum.
Apologies if i have misunderstood your problem.