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.