$$\sum_{i=a}^b f(i)g(i)$$
Where $f$ and $g$ aren't constant. I know this might be very broad, but all I'm looking for is to be pointed in the direction of theorems/disciplines that deal with this. Anything, like a rule that states I can move the components around in a specific way, which might be a more helpful reformulation.
In my case, I know what $\displaystyle \sum_{i=a}^b f(i)$ is, and I know what $\displaystyle \sum_{i=a}^b g(i)$ is. A rule that relates their respective values to the value of the above formula would definitively provide a solution for me.