Question about sums with a negative limit for the index

Question

To me, it looks like we have $\;\sum_{i = 1}^{0} x_i = 0\;$ and $\;\sum_{i = 1}^{1} x_i = x_1\;$. What happens if I write the following? $$\;\sum_{i = 1}^{-123} x_i\;$$ Would this be defined?

Answer 1

There are different conventions:

• the sum is $0$;

• the expression is undefined;

• the expression is interpreted as $\displaystyle\sum_{i=-123}^1x_i$.

In my experience the first is the most common, and the last is quite uncommon.