geometric series

A geometric series is the sum you get when each term is a fixed multiple of the one before it — you keep scaling by the same factor over and over. If that factor shrinks every term toward zero, the endless sum settles down; if it does not, the sum runs away.

Concretely, a geometric series has the form a + a r + a r^2 + a r^3 + ... = sum a r^n, where a is the first term and r is the common ratio. Its partial sum has the closed form s_N = a (1 - r^{N+1}) / (1 - r) for r not equal to 1. As N -> infinity, r^{N+1} -> 0 exactly when |r| < 1, giving the celebrated formula sum_{n=0}^infinity a r^n = a / (1 - r) for |r| < 1. If |r| >= 1 (and a is nonzero) the series diverges.

The geometric series is the workhorse of convergence theory: it is the standard against which the comparison test, ratio test, and root test implicitly measure other series, and it is one of the few series whose exact sum is available in closed form. Its quick decay when |r| < 1 also underlies error estimates for many approximations.