If a series converges does its sequence of partial sums converge?

Question

By Definition, a series $\displaystyle \sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $\displaystyle S_n = \sum_{k=1}^n a_k$ converges.

My question is, is the converse true? If $\displaystyle \sum_{n=1}^\infty a_n$ converges does it mean $\{S_n\}$ converges?

Answer 1

This community wiki solution is intended to clear the question from the unanswered queue.

---

We define convergence of a series as follows:

The series $\displaystyle \sum_{k = 1}^\infty a_k$ converges if and only if its sequence of partial sums $\displaystyle S_n = \sum_{k = 1}^n a_k$ converges.

When stating definitions, authors write "if" instead of "if and only if" as mentioned in the comments.