 Here is an infinite geometric sequence: 16, 8, 4, 2, ... t1 = 16 r = ½ You know how to find the sum of any number of terms For example, the sum of the first six terms: Sn = t1(rn - 1) (r - 1) S6 = 16(½6 - 1) (½ - 1) S6 = 31.5 Suppose however we want to find the sum of the entire infinite sequence. Can we find the sum of an infinite number of things? Let's try! Here's the same sequence extended out to twelve terms: 16, 8, 4, 2, 1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, .... Let's start adding terms: S1 = 16 S2 = 16 + 8 = 24 S3 = 24 + 4 = 28 S4 = 28 + 2 = 30 S5 = 30 + 1 = 31 S6 = 31 + 0.5 = 31.5 S7 = 31.5 + 0.25 = 31.75 S8 = 31.75 + 0.125 = 31.875 S9 = 31.875 + 0.0625 = 31.9375 S10 = 31.9375 + 0.03125 = 31.96875 S11 = 31.96875 + 0.015625 = 31.984375 S12 = 31.984375 + 0.0078125 = 31.9921875 We're still a long way from an infinite number of terms. However, it sort of looks like we're not going to reach a sum of 32: It seems that the amount we're adding each time is less than the distance to 32 which means the sum is going to get closer and closer to 32 but won't ever reach it unless we add an infinite number of terms! The sum is getting closer and closer to 32, which it will reach if there are an infinite number of terms We can say this using limits: limn→∞Sn = 32 In other words: S∞ = 32 The reason that we were able to find a sum of an infinite number of terms is because the value of r was a fraction whose size was less than 1 Negative values for r would work too, as long as the size of r is less than 1 Using absolute value notation: |r| < 1 -1 < r < 1 |