calculus - About the definition of convergent series meaning.

From my textbook, the definition of convergent series is given as follow.

If the sequence of partial sums {$S_n$} is convergent and $\displaystyle{\lim_{n \to \infty}} S_n$ exists, then the series $\sum a_n$ is called convergent.

So if I know that $\sum a_n$ is convergent, can I say that {$S_n$} is convergent and $\displaystyle{\lim_{n \to \infty}} S_n$ exists? As far as I know the answer is yes, but why is "if then" statement used here in the definition.