If we are given a function, $f(x)$, we can either integrate it or sum it. I'm wondering what integration can do with $f(x)$ that summation can't, and what summation can do that integration can't.
For example, we can make a list of what is easy for integration, and what is easy for summation:
$$ \begin{array}{|c|c|} \hline \text{Integration} & \text{Summation} \\ \hline \frac{1}{f(x)} & \binom{x}{n} \\ f(x) \to f(ax+b) & \text{falling factorials} \\ \dots & \dots \\ \hline \end{array} $$
I'm wondering if someone can give us a general feel for what the pros and cons are of integration versus summation. In other words, what functions are easier for one or the other?