Special Case of Summation

Question

Hello what would be the solution to the summation over the range from 1 to 0?

$$ \sum_{1}^{0} = ? $$

My guess is -1 or 0, but I can't find any reference to this case.

Answer 1

We can use the inductive identity

$$\sum_{i=m}^{n+1}a_i=\sum_{i=m}^{n}a_i+a_{n+1}$$

to get

$$\sum_{i=1}^0 a_i=\sum_{i=1}^1 a_i-a_1=a_1-a_1=0$$

Of course, that assumes the usual rules apply to such a case. That is a weak assumption, so such sums as yours should be avoided.

Answer 2

If $m > n$, then: $$ \sum\limits_{k=m}^{n} a_k = 0. $$

Answer 3

Summation of what? Anyway, whatever it is, the answer is $0$. Because the sum is not well-defined, so it will not be performed.