Series, approximation & Big-O

The idea

A series adds a sequence of terms one after another; convergence asks whether the running total settles toward a fixed number as more terms are added. Finite partial sums let computers approximate genuinely infinite processes to any precision needed. The same instinct — ignoring small details to focus on how a quantity scales — shows up in Big-O growth classes for algorithm analysis.

Real-world: downloading a file in shrinking chunks

Imagine receiving half a file, then half of what remains, and so on: $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots$. Each chunk is smaller, but the running total approaches one complete file. After three chunks, $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} = \tfrac{7}{8} = 87.5\%$ has arrived — the infinite series converges to 1, not infinity.

Worked example

1. Start with $1$
2. Add $\tfrac{1}{2}$: total is $1.5$
3. Add $\tfrac{1}{4}$: total is $1.75$
4. The remaining gap keeps halving, so the total approaches $2$

Cheat sheet

$1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots = 2$

• A geometric series $a + ar + ar^2 + \cdots$ converges when $|r| < 1$, to $\dfrac{a}{1-r}$.
• Partial sums are the finite approximations a computer actually calculates.
• Big-O ignores constants and low-order terms to describe how cost scales for large $n$.
• A series can have infinitely many nonzero terms and still sum to a finite number, as long as the terms shrink fast enough.