Is $\sum_{n=0}^\infty (a \cdot r^n)$ equivalent to $\lim_{n \to \infty}\sum_{k=0}^n (a \cdot r^k)$? In other words, when writing down an infinite sum, are we always implying that it's actually the limit of that series as the number of terms approaches infinity, or is there some subtle difference?

The sum
$$\sum_{n=0}^\infty$$
is defined to be
$$\lim_{k\to\infty}\sum_{n=0}^k$$
so yes they are the same. Of course this is an abuse of notation, since $\infty$ is not a number. In the same way, it is not "proper" to write the interval $[0,\infty)$, but we all understand what it means.
