Final answer:
The series ∑n=1∞(2n+3n)/6n converges because it is the sum of two geometric series with common ratios less than 1. The sum of the series is 5/6.
Step-by-step explanation:
The question asks to determine whether the series ∑n=1∞(2n+3n)/6n converges or diverges and to find its sum if it converges. We can consider this series as a sum of two separate series: ∑n=1∞(2n/6n) and ∑n=1∞(3n/6n). Simplifying, we get two geometric series: ∑n=1∞(1/3)n and ∑n=1∞(1/2)n, both of which are known to converge because their common ratios (1/3 and 1/2) are less than 1.
The sum of the first series is S1 = a/(1-r) = 1/2, and for the second series, it's S2 = a/(1-r) = 1/3. So, the sum of the original series is Stotal = S1 + S2 = 1/2 + 1/3 = (3+2)/6 = 5/6.