Confusion with Summations I am having a little bit of confusion regarding summations.
I know that $$\sum_{i=m}^n a_i = a_{m}+a_{m+1}+\cdots +a_{n-1}+a_n$$
Here is my confusion. How do we interpret/decompose the following: 
$$ \sum_{i=m}^n a_i~~~~,~~~m=0,1,2,3,,,,k~~?  $$

Is it the $m=0,1,2,3,\dots,k$ that's confusing you?  That just means there are several sums: $$
\text{$\sum_{i=0}^na_i$, $\ \ \sum_{i=1}^na_i$, $\ \ \sum_{i=2}^na_i$, $\ \ \dots\ $, $\ \sum_{i=k}^na_i$}$$
