Infinite decreasing progression

A geometric progression whose common ratio has modulus less than one, i.e. |q| < 1, is called infinite decreasing; its terms get closer to zero as n grows. If n grows without bound, q^n tends to zero, which we write lim q^n = 0 (n → ∞). So in S_n = b₁ : (1 − q) − b₁q^n : (1 − q) the second part tends to zero, while the first does not depend on n. The sum of an infinite decreasing progression is the limit of S_n as n → ∞, and it equals S = b₁ : (1 − q). This formula is valid only for |q| < 1. We also use it to turn a periodic decimal into an ordinary fraction: 0.(7) = 7/10 + 7/100 + ... .