I understand, generally that the forces determining whether an infinite series will converge or diverge are the rate at which the terms in the series i.e. $a_n$ decrease or increase and the rate at which you add each term. Is there a way to discuss these "forces" in a more precise way or at least a way to explore them more deeply?
EDIT: I understand how to determine and prove that a series will converge or diverge, my question lies more along the lines of what would cause an infinite series to converge.
There are many convergence tests that address the properties you bring up. One that comes to mind is the $p$ test, which states the following:
Consider the infinite series $\sum\frac{1}{n^p}$. If $p \leq 1$, the series diverges. Else, it converges.
There are many more, but you've probably seen this $p$ test if you've taken Calculus II.