Partial Sum
A partial sum is the sum of a finite number of terms from the beginning of a sequence, used to study infinite series.
Formula
S_n = \sum_{k=1}^{n} a_k
Definition
A partial sum is the sum of just the first few terms of a sequence, not the entire infinite sequence, you stop after $n$ terms. The $n$-th partial sum is $S_n = a_1 + a_2 + \ldots + a_n = \sum_{k=1}^{n} a_k$, and an infinite series converges if the sequence of partial sums $\{S_n\}$ has a limit $L$, in which case $\sum_{k=1}^{\infty} a_k = L$. The Cauchy criterion formalizes this: given a series $\sum a_n$ in a topological space, it converges if and only if $\{S_n\}$ is a convergent (equivalently, Cauchy) sequence, meaning for all $\epsilon > 0$ there exists $N$ such that $|S_m - S_n| < \epsilon$ for all $m, n > N$.
Example
For $1, 1/2, 1/4, 1/8, \ldots$: $S_1 = 1$, $S_2 = 1.5$, $S_3 = 1.75$, $S_4 = 1.875$, each partial sum getting closer to $2$. For the geometric series $\sum 2^{-k}$ from $k=1$, $S_n = 1 - (1/2)^n$, so $S_n \to 1$ as $n \to \infty$, meaning the infinite sum equals $1$. By contrast, the harmonic series $\sum 1/n$ diverges despite $a_n \to 0$: grouping terms as $1 + (1/2) + (1/3+1/4) + (1/5+\ldots+1/8) + \ldots$ shows each group exceeds $1/2$, so the partial sums grow without bound.
Key Insight
Partial sums are how we make sense of adding infinitely many numbers: we watch what happens as we add more and more terms, and if they approach a fixed number, the infinite series converges to that number. The distinction between a series (an expression indicating addition) and its partial sums (the actual numerical values) is subtle but crucial, a series is shorthand, its value is defined as the limit of partial sums. Studying partial sums as a sequence in a complete metric space (Banach space) gives the framework for absolute convergence, conditional convergence, and the various convergence tests: comparison, ratio, root, and integral.