Partial sum ($S_n$)

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 areBy 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.

For a convergent series with sum $S$ and $N$-th partial sum $S_N$, its series remainder is $R_N=S-S_N$. It equals the sum of the terms after position $N$. Its absolute value measures the error of replacing $S$ by $S_N$; it need not be positive when the series has signed sequence terms.