Final answer:
To find the positive values of b for which the series converges, we need to determine when the series satisfies the convergence condition. The positive values of b for which the series converges are all numbers greater than 1. This can be represented in interval notation as (1, ∞).
Step-by-step explanation:
A series converges when the limit of its terms approaches a finite value as the number of terms increases. In this case, we have the series Σ(1/b)^n where n ranges from 1 to infinity. To find the positive values of b for which the series converges, we need to determine when the series satisfies the convergence condition.
To do this, we use the geometric series formula: Σ ar^(n-1) = a / (1 - r), where a is the first term and r is the common ratio. Comparing this formula to our series, we have a = 1 and r = 1/b.
For the series to converge, the absolute value of the common ratio (1/b) must be less than 1. So, we have |1/b| < 1, which simplifies to 1/b < 1. Solving this inequality, we find that b > 1. Therefore, the positive values of b for which the series converges are all numbers greater than 1, which can be represented in interval notation as (1, ∞).