= Partial sum
{title2=$S_n$}
{wiki}
The $N$th <partial sum> of a <series> with terms $(a_n)$ is the finite <sum> $S_N=\sum_{n=1}^N a_n$. It records the accumulated total after $N$ additions. The terms and their <partial sums> are different <sequences>: the terms of the <harmonic series> get small, while their <partial sums> grow without bound.
For the <telescoping series> with terms $a_n=1/[n(n+1)]$, the first three <partial sums> are
$$
S_1=\frac12,\qquad S_2=\frac12+\frac16=\frac23,\qquad
S_3=\frac12+\frac16+\frac1{12}=\frac34.
$$
By contrast, the <harmonic series> gives $H_1=1$, $H_2=3/2$, and $H_3=11/6$, where $H_N$ is the <harmonic number>. Computing more <partial sums> approximates an infinite sum only when a <limit of a sequence> exists, and the <tail of a telescoping reciprocal series> shows how an error estimate can certify the approximation.
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