Mathematics
Partial Sums
Quick fact
The concept of partial sums was used by Archimedes over 2,000 years ago to approximate π by adding areas of polygons.
Why this is interesting
You've probably added up a list of numbers before—but what if you only add the first few and ignore the rest? That simple act is the key to understanding infinite sums.
Read the full explanation
Understanding Partial Sums
Consider a list of numbers: 1, 2, 3, 4, 5, ... Adding the first three gives 1+2+3=6—that's a partial sum, often written as S₃. In general, the partial sum Sn is the sum of the first n terms of a sequence. If the sequence is a pattern like 1, ½, ¼, ... (a geometric series), then S₁=1, S₂=1.5, S₃=1.75, and so on. Notice how these sums get closer to 2. This 'getting closer' is the seed for the idea of convergence. Partial sums are like snapshots of a process that might have a final destination.
A deeper explanation
Partial sums reveal whether an infinite series adds up to a finite number (converges) or grows without bound (diverges). For a series like 1 + ½ + ¼ + ..., each partial sum is just a little closer to 2 than the previous one. The limit of the sequence of partial sums (as n→∞) defines the sum of the infinite series. This idea underpins calculus (e.g., Taylor series represent functions as infinite sums of polynomials) and practical calculations like compound interest, where partial totals show growth over time. Without partial sums, infinite processes would remain a mystery; with them, we can precisely define and compute quantities that are otherwise infinite.