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jimmyrepinIt will converge if the common factor r is between 0 and 1. For example:- A GS with first term a1 = 2 and d = 1/2, the terms are:- 2 , 1 , 1/2, 1/4, 1/8,1/16 and so on Each term gets smaller and smaller and they will become very small - close to 0. The sum will converge to a certain value also. In this case it will converge to 2 /( 1 - 1/2) = 4 If d > 1 the terms will increase without bounds and the sum gets bigger and bigger . The series diverges. for example 2, 8, 32, 128, 512 .... where a1 = 2 and d = 4.