# Definition:Series/Sequence of Partial Sums

## Definition

Let $\sequence {a_n}$ be a sequence in a structure $S$.

Let $s$ be the the series:

$\displaystyle s = \sum_{n \mathop = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$

The sequence $\sequence {s_N}$ defined as the indexed summation:

$\displaystyle s_N: = \sum_{n \mathop = 1}^N a_n = a_1 + a_2 + a_3 + \cdots + a_N$

is the sequence of partial sums of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.