A little curiosity about sum notation

Question

In summation notation($\sum$), can the stopping point be smaller than the starting point?
For example, can I say $$\sum_{i=1}^0 i = 0$$
because $\ 1 > 0$ so it does not sum anything??

Answer 1

Bear in mind $\sum_{i=a}^bf(i)$ is shorthand for $\sum_{i\in S}f(i)$ with $S:=\{i\in\Bbb Z|a\le i\le b\}$, so your notation is a special case of an empty sum, as would be $\sum_{i=1}^{-1}i$. In general, $\sum_{i=a}^bf(i)$ sums $\max(b-a+1,\,0)$ elements (you can also denote this $(b-a+1)^+$). You can get empty products with the same rules, just replacing $\sum$ with $\prod$.

Answer 2

Wikipedia gives the following formal definition:

$$\begin{align} \sum_{i=a}^bg(i)&=0,\text{ for }b\lt a\\ \sum_{i=a}^bg(i)&=g(b)+\sum_{i=a}^{b-1}g(i),\text{ for }b\ge a. \end{align}$$

Remark: It's somewhat unclear (to me at least) whether Wikipedia is tacitly assuming that $a,b\in\mathbb{Z}$ in its recursive definition. This could be important because you sometimes see people write things like

$$\sum_{i=0}^\sqrt ng(i)$$

where they (probably) mean

$$\sum_{i=0}^{\lfloor\sqrt n\rfloor}g(i)$$