Let $$m, n$$ be integers, and let $$(a_i)_{i=n}^{m}$$ be a finite sequence of real numbers, assigning a real number $$a_i$$ for each integer $$i$$ between $$n$$ and $$m$$ inclusive (i.e. $$m \le i \le n$$). Then we define the finite sum (or finite series) $$\sum_{i=m}^{n} a_i$$ by the recursive formula
\begin{aligned}\sum_{i=m}^{n} a_i &= 0 \text{ whenever } n < m; \\\sum_{i=m}^{n+1} a_i &= [...] \text{ whenever } [...].\end{aligned}