Geometric Series
A constant ratio sums to a clean closed form — if |r| < 1.
Concept
Shrinking terms, finite total
Each term is r times the last. When |r| < 1 the terms shrink fast enough that the infinite sum converges to a/(1 − r).
When |r| ≥ 1 the terms do not go to zero, so the series diverges — no closed form to find.
∑ ar^n = a / (1 − r), |r| < 1