Partial Sum
Functions & Advanced AlgebraA 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.
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 adds one more term. They keep getting closer to $2$.
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. If the partial sums approach a fixed number, the infinite series converges to that number.
Definition
The $n$-th partial sum of a series is $S_n = a_1 + a_2 + \ldots + a_n = \sum_{k=1}^{n} a_k$. An infinite series converges if the sequence of partial sums $\{S_n\}$ has a limit $L$: then $\sum_{k=1}^{\infty} a_k = L$.
Example
Geometric series $\sum 2^{-k}$ from $k=1$: $S_1 = 0.5$, $S_2 = 0.75$, $S_3 = 0.875$, $\ldots$, $S_n = 1 - (1/2)^n$. As $n \to \infty$, $S_n \to 1$. So the infinite sum equals $1$.
Key Insight
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.
Definition
Given a series $\sum a_n$, the sequence of partial sums $\{S_n\}$ is defined by $S_n = \sum_{k=1}^{n} a_k$. The series converges iff $\{S_n\}$ is a convergent sequence in the underlying topological space. The Cauchy criterion for series: $\sum a_n$ converges iff for all $\epsilon > 0$, there exists $N$ such that $|S_m - S_n| < \epsilon$ for all $m, n > N$.
Example
The harmonic series $\sum 1/n$ diverges despite $a_n \to 0$: the partial sums grow without bound ($S_{2^n} > 1 + n/2$ by grouping). This shows that $a_n \to 0$ is necessary but not sufficient for convergence.
Key Insight
Studying partial sums as a sequence in a complete metric space (Banach space) gives the framework for absolute convergence, conditional convergence, and various convergence tests (comparison, ratio, root, integral).