For some sequences \((a_n)_{n\in\mathbb N}\) the sequence of partial sums\((s_n)_{n\in\mathbb N}\) defined by \(\textstyle s_n = \sum_{k=1}^\infty a_k\) converges. The limit of this sequence is denoted by \(\textstyle \sum_{n=1}^\infty a_n\) .

The term "formal sum" here refers to the fact that the sequence \((s_n)_{n\in\mathbb N}\) does not always converge. However, we still write \(\textstyle \sum_{n=1}^\infty a_n\), just like we write \(\lim a_n\) write for sequences that do not necessarily converge.