geometric series
A geometric series adds up a list of numbers where each one is the previous one multiplied by the same fixed factor. Start with some amount, then keep taking the same fraction of what you had: a coin's bounce reaching the same percentage of its previous height each time, or a drug dose halving in the body every few hours. If that factor shrinks the terms, the running total tops out at a finite value.
It has the form a + a*r + a*r^2 + a*r^3 + ... where a is the starting term and r is the common ratio. When |r| < 1 the series converges, and there is a clean closed-form sum: a + a*r + a*r^2 + ... = a/(1 - r). When |r| >= 1 the terms do not shrink toward 0, so the series diverges. The partial-sum formula a*(1 - r^n)/(1 - r) makes this transparent: as n -> infinity, r^n -> 0 exactly when |r| < 1.
The geometric series is the cleanest example of an infinite sum with a finite total, and it is the workhorse behind many results. The repeating decimal 0.999... is the geometric series 9/10 + 9/100 + 9/1000 + ..., which sums to exactly 1 — that is why 0.999... and 1 are the same number, not merely close.
With a = 1/2 and r = 1/2 this gives (1/2)/(1 - 1/2) = 1, the half-distance walk.
The closed form a/(1 - r) holds only when |r| < 1; if |r| >= 1 the series diverges and that formula does not apply.