Math and science::Analysis::Tao::07. Series
Finite series, definition
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\[\begin{aligned}\sum_{i=m}^{n} a_i &= 0 \text{ whenever } n < m; \\\sum_{i=m}^{n+1} a_i &= [...] \text{ whenever } [...].\end{aligned}\]