Given a geometric progression { a_n } = a₁, a₂, a₃, ...., a_n , ...., if the absolute value of the common ratio, | r | , is less than 1, the corresponding geometric series S_n = a₁ + a₂ + a₃ + .... + a_n + ... will converge to a finite value as n approaches infinity. This value can be calculated as follows:

S_¥ = a₁ / (1 - r) where

a₁ is the first term of the geometric progression
r is the common ratio of the geometric progression, | r | < 1
S_¥ indicates that the series sums up all the infinite terms in the
sequence.

Note that when | r | > 1, the terms in the geometric sequence will get progressively larger, approaching infinity as n goes to infinity. Therefore, the corresponding geometric series will not converge. So, as a warning, the above formula must not be used when | r | > 1.

EX. The sequence { a_n } = 1, 1/2, 1/4, 1/8, .... is a geometric progression with a₁ = 1 and r = 0.5. Since | r | = | 0.5 | < 1, the sum of all the infinite terms in this sequence will converge to a finite number. That number is

S_¥ = 1 / (1 - 0.5) = 1 / 0.5 = 2