Final answer:
The sum of the convergent series ∑_n=3^∞ 2(1/3)^(n-1) is 2/3.
Step-by-step explanation:
To find the sum of the convergent series ∑n=3∞ 2(1/3)n-1, we can identify that the expression inside the summation is a geometric series with a common ratio of 1/3. A geometric series converges when the absolute value of the common ratio is less than 1. In this case, the sum converges because |1/3| = 1/3, which is less than 1.
To find the sum, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
Plugging in the values, we get S = 2(1/3) / (1 - 1/3) = 6/9 = 2/3.