Does the infinite geometric series diverge or converge? Explain. 1/5 + 1/10 + 1/20 + 1/40 + .... (1 point)
Responses
It diverges; it has a sum.
It diverges; it does not have a sum.
It converges; it has a sum.
It converges; it does not have a sum.
It converges; it has a sum.
This geometric series has a common ratio of 1/2, which is less than 1. Therefore, the series converges. The sum of an infinite geometric series with a common ratio |r| < 1 is S = a / (1 - r), where a is the first term of the series. In this case, the first term a = 1/5, and the common ratio r = 1/2. So the sum of the series is S = (1/5) / (1 - 1/2) = (1/5) / (1/2) = 2/5. Therefore, the series converges to 2/5.