Given a geometric progression { an } = a1, a2, a3, ...., an , ...., if the absolute value of the common ratio, | r | , is less than 1, the corresponding geometric series Sn = a1 + a2 + a3 + .... + an + ... 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 { an } = 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
