Sum with conflicting specifier

Question

This is the fomula (there might be some terms added, it is just boiled down to these two sums): $$\sum_{i = 0}^{N} \Big(\ Term_i * \sum_{j = 0}^{i - 1} x_j\ \Big)$$ The outer sum will iterate from $0$ to $N$ and the inner one from $0$ to $i - 1$. It has a recursive character. Is the inner sum valid on the first term ($i = 0 $ and $j = -1$)?

Or do I have to rewrite the formula this way: $$Term_0 + \sum_{i = 1}^{N} \Big(\ Term_i * \sum_{j = 0}^{i - 1} x_j\ \Big)$$

This is all about how to write down the formula in a correct manner.

Answer 1

When $i=0$, the inner sum is taken over all $j$ that satisfy $j\ge 0$ and, simultaneously, $j\le-1$. There are no such $j$, so the sum is over the empty set. The empty sum contributes nothing, i.e., $0$. -- Gerry Myerson (slightly edited)