Partial sums - Knowlpedia

For a sequence $(a_k)_{k\ge1}$ of real or complex numbers, the $n$ th partial sum is the finite sum $s_n=\sum_{k=1}^n a_k$ , for $n\ge1$ . Equivalently, $s_0=0$ and $s_n=s_{n-1}+a_n$ .

Relation to a series

The sequence $(s_n)$ encodes the series $\sum_{k=1}^{\infty}a_k$ : statements about convergence or divergence of the series are statements about whether the partial sums form a sequence with a limit, as formalized in convergent series.

Examples

For $a_k=\frac{1}{k}$ , the partial sums are $s_n=\sum_{k=1}^n \frac{1}{k}$ (harmonic numbers).
For $a_k=r^{k-1}$ , the partial sums are $s_n=\sum_{k=1}^n r^{k-1}=\frac{1-r^n}{1-r}$ (when $r\neq 1$ ).