What is the difference between the summations?

Question

What is the difference between the summation $$\sum_{1 \leq i<j \leq n} f(i,j)$$ and $$\sum_{1\leq i} \sum_{<j \leq n} f(i,j)?$$

Answer 1

The second is ambiguous. $\sum$ usually means you will iterate over some index individually. A double sum implies summing over two indices individually.

The first version you give implies summing with respect to $i,j$ in a single intration, whereas the second vaguely implies two.

So I'd say the first is the better of the two, the second having ambiguous meaning.

Answer 2

The second one is nonsense; the condition in the sum, $<j\leq n$, doesn't mean anything.