Infinite Geometric Series

Adding infinitely many terms sometimes gives a finite total.

The test is the common ratio. If |r| < 1, the terms shrink fast enough that the sum converges:

S = a₁/(1 − r)

For 8 + 4 + 2 + 1 + …: a₁ = 8 and r = 1/2, so S = 8/(1 − 1/2) = 16. It never exceeds 16, no matter how many terms you add.

If |r| ≥ 1, the terms do not shrink and the sum grows without bound. There is no finite answer, and the formula does not apply.

This is also the clean proof that 0.999… = 1: it is 0.9 + 0.09 + 0.009 + …, with a₁ = 0.9 and r = 0.1, giving 0.9/0.9 = 1.

The infinite geometric series Σ_{k=0}^{∞} a₁r^k converges precisely when |r| < 1, with sum S = a₁/(1 − r). This follows from the finite sum Sₙ = a₁(1 − rⁿ)/(1 − r) by taking the limit: rⁿ → 0 exactly when |r| < 1.

For |r| ≥ 1 the partial sums do not approach a limit and the series diverges — including the case r = −1, where the partial sums oscillate between two values without settling.

The result gives the standard conversion of repeating decimals to fractions, and it is the first example students meet of an infinite process with a finite result, which is the conceptual groundwork for limits.

Applications include the total distance travelled by a bouncing ball with a constant rebound ratio, and the economic multiplier, where each round of spending is a fixed fraction of the previous one and the total is finite despite the process continuing indefinitely.