Summation and Product notation: Convention for when final index number is smaller than the first?

Question

What, for example, are the following?

1. $\sum_{i=5}^4i$
2. $\prod_{i=5}^4i$

Is there a standard convention for what they should be? My guess is that $\sum_{i=5}^4i=0$ and $\prod_{i=5}^4i=1$. Is that correct?

Answer 1

The notation $$\sum_{i=j}^k a_i$$ is really shorthand for $$\sum_{i\in S}a_i $$ where $$ S = \{i\in\mathbb Z: j\leqslant i\leqslant k\}. $$ So if for example $j=5$, $k=4$, we have $$\sum_{i=5}^4 a_i = \sum_{i\in\varnothing} a_i = 0, $$ as the empty sum is generally accepted to be $0$ by convention. The product would be $$\prod_{i=5}^4 a_i = \prod_{i\in\varnothing} a_i = 1, $$ as the empty product is $1$ by convention.

Answer 2

The notation $\sum_{i=a}^{b}f_i$ means $\sum_i f_i[a\leq i\leq b]$, where $[\cdot]$ is the Iverson symbol/bracket which is one when the proposition inside is true and zero otherwise.

This interpretation would allow you to define the summation symbol even when $b<a$. In that case the inequalities inside the Iverson symbol are never true. So, all the terms in the summation are zero.

On the other hand, this is just a convention. In some contexts it is useful to consider other options. For example, define $\sum_{i=5}^{4}f_i=-\sum_{i=4}^{5}f_i$ when the sum is being seen as an integral $\int_{5}^{4}f_id\mu(i)=-\int_{4}^{5}f_id\mu(i)=-\sum_{i=4}^{5}f_i$ in the counting measure of the integers.