Definition (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then we define the finite sum $\sum_{i=m}^{n} a_i$ by the recursive formula $\sum_{i=m}^{n} a_i = 0 $ whenever $n < m$ and $\sum_{i=m}^{n+1} a_i = \sum_{i=m}^{n} a_i + a_{n+1}$ whenever $ n \geq m-1$ .
Lemma Let $m \leq n < p$ be integers, and let $a_i$ be a real number assigned to each integer $m \leq i \leq p$. Then we have $\sum_{i=m}^{n} a_i + \sum_{i=n+1}^{p} a_i =\sum_{i=m}^{p} a_i$.
MY QUESTIONS
- What are the typical questions one would ask of himself in trying to understand such a typical sequence of definition and lemma?
- Why did the author write $m \leq i \leq p$ and not $(n+1) \leq i \leq p$? Is the author trying to generalize $i$? If yes what is the benefit of this generalization?