AHL 1.11 (HL)—Infinite geometric sequences

For a geometric series with first term a and common ratio r, the infinite sum is S∞ = a/(1 − r) only when |r| < 1. That condition means successive terms approach zero, so adding more terms changes the total by smaller and smaller amounts.

The series 12 + 6 + 3 + … has a = 12 and r = 1/2, so S∞ = 12/(1 − 1/2) = 24. By contrast, 12 + 24 + 48 + … has |r| > 1 and diverges.

Do not use the formula merely because a series is geometric. If r = 1 the terms never decrease, and if |r| > 1 they grow or fail to vanish; neither case has a finite sum to infinity.