What is series?

Example 17.2 (Geometric series)

For $q \in \C$: $\;S_N = \sum_{n=0}^{N} q^n = \frac{1 - q^{N+1}}{1-q}$ ($q \neq 1$). The series converges iff $\abs q < 1$ (Exercise 11.3), with

Example 17.3 (Periodic decimals are geometric series)

What number is $0.363636\dots$? Its very writing is a series:

by the geometric sum with $q = \frac{1}{100}$. In general a block $B$ of $p$ digits repeating forever is worth $\frac{B}{10^p - 1}$ — the mechanism behind the periodicity criterion of Problem 10.1, which this chapter’s language finally states in one line: a decimal expansion is a convergent series, eventually periodic exactly when its sum is rational. The digit machinery of Chapter 10, built there with bare suprema, was series theory traveling incognito.

Example 17.5 (Planning digits with the geometric remainder)

For $\abs q < 1$ the remainder of the geometric series is explicit:

This converts accuracy goals into term counts before any computation. To evaluate $\sum_{n\geq0} \bigl(\frac13\bigr)^n = \frac32$ within $10^{-10}$: need $\frac{(1/3)^{N+1}}{2/3} \leq 10^{-10}$, i.e. $3^{N} \geq \frac{3}{2}\cdot 10^{10}$, i.e. $N \geq 22$ (as $3^{22} \approx 3.1\cdot10^{10}$): twenty-three terms, known in advance. Every geometric-rate estimate of the weekend problems (the $\frac13$-series for $\ln 2$, Machin’s arctangents in Problem 16.1) is this two-line budget in professional dress.