Sequences, series, and the arithmetic of infinity
A sequence is an ordered list governed by a rule: a₁, a₂, a₃, ... The rule may be explicit (aₙ = 2ⁿ) or recursive, each term defined from earlier ones. The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, ... — is the most famous recursive example, each term the sum of the two before it, and it appears whenever growth feeds on its own backlog: branching populations, spiralling shells, compound interest intervals.
A series is the sum of a sequence's terms. The philosophical shock is that an infinite list can have a finite total. The geometric series 1 + 1/2 + 1/4 + 1/8 + ... adds each term toward — but never past — 2, and the general formula a/(1 - r) captures this whenever the ratio r has magnitude below 1. Above 1 the terms grow and the sum diverges. That threshold, |r| < 1, decides convergence, and the same boundary appears in power series, probability generating functions, and the radius of convergence of every Taylor expansion.
Growth hierarchies matter in combinatorics. A factorial n! counts orderings of n distinct objects and rockets upward: 10! = 3,628,800 while 20! ≈ 2.4 x 10¹⁸. Combinations (choosing without order) divide the ordering count by k! to correct for overcounting, which is why "n choose k" stays finite and symmetric — n choose k equals n choose (n - k).