Partial sums

A partial sum is a finite sum $s_n=\sum_{k=1}^n a_k$ associated to a series $\sum_{k=1}^\infty a_k$.

The sequence $(s_n)$ encodes the series: 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 $r\neq 1$).