"$n^{th}$ partial sum" Does the phrase, "$n^{th}$ partial sum" refer to $\sum_{k=0}^{n-1} f(k)$ or $\sum_{k=0}^n f(k)$?
 A: The correct answer to your question is "Yes." :P
A more useful answer is that either sum could reasonably be called the "$n$-th partial sum"—this phrase is more descriptive than technical, and it is does not have a universally agreed upon technical definition.  Some authors prefer to start counting at zero; others at one.  Indeed any one of
$$ \sum_{k=0}^{n-1} f(k), \qquad
\sum_{k=0}^{n} f(k), \qquad\text{or}\qquad
\sum_{k=1}^{n} f(k) $$
could reasonably be called the $n$-th partial sum.  Moreover, the distinction between these sums is rarely terribly important or significant—the precise form of the $n$-th partial sum is usually far less important than the limit.  That being said, if I were forced to pick a side, my own preference would be to call the latter sum the "$n$-th partial sum".
Of course, if it really does become necessary to work with the partial sums directly, one should make clear exactly what is meant.  For example, write

Definition:  Let $f : \mathbb{N} \to \mathbb{R}$, and define
  $$S(n) := \sum_{k=0}^{n} f(k). $$
  The value $S(n)$ is the $n$-th partial sum.

Finally, I would like to reiterate JMoravitz's comment:  it can never hurt to ask for clarification.  If you see this in lecture notes or on an exam, it is reasonable to politely ask the instructor or proctor for clarification.  In this setting the goal is to learn—if confusion about notation is preventing your from learning, or preventing you from demonstrating your understanding, then you should absolutely ask for a more precise statement.
