I read this related thread but it doesn't give me a satisfactory answer to the following question:
Must it be true that the order of indices in a sum is relevant? A finite sum is essentially adding up all elements in a set. Who cares it that set is ordered forwards or backwards? Consider the following examples.
Common convention tells us that
$$\sum_{n=5}^0n=0$$
even though we are essentially trying to add the elements of {$n\in[0,5]$} backwards. The validity of this becomes paramount in the following case,
$$\sum_{m=0}^n f(n-m)\ne0$$
when we make a simplifying change of variable $m=n-i$. Then, the sum becomes
$$\sum_{i=n}^0 f(i)=0, \text{by convention.}$$
Obviously nothing significant has changed with the sum when we just change a variable definition, Why, then, does the convention dictate that the value of the sum must change? It seems this particular sum convention is nonsense and should be abandoned.
If you write the summation as $$ \sum_{0\le m\le n}f(n-m) $$ and do the substitution $m=n-i$, then the condition in the sum becomes $$ 0\le n-i\le n $$ which is equivalent to $$ n\ge i\ge 0 $$ which in turn can be written as $0\le i\le n$.
Therefore $$ \sum_{0\le m\le n}f(n-m)=\sum_{0\le i\le n}f(i) $$
A summation such as $$ \sum_{n=5}^0 f(n) $$ can (should?) be written as $$ \sum_{5\le n\le0}f(n) $$ and no index $n$ satisfies the condition, so the summation is by definition $0$.