Final answer:
The series Σₙ ₌ ₁ (bₙ) ¹/ⁿ does not converge.
Step-by-step explanation:
To determine if the series Σₙ ₌ ₁ (bₙ) ¹/ⁿ converges, we need to consider the limit of (bₙ)¹/ⁿ as n approaches infinity. Since the sequence (bₙ)ₙ = 1∞ converges to 2/3, we can use this information to determine the convergence of the series.
As n approaches infinity, the exponent ¹/ⁿ approaches 0. Therefore, we can rewrite the series as Σₙ ₌ ₁ (bₙ)⁰ = Σₙ ₌ ₁ 1, which is an infinite geometric series with a common ratio of 1 and a first term of 1.
The sum of an infinite geometric series is only finite if the common ratio is less than 1. In this case, the common ratio is 1, so the series does not converge. Therefore, the series Σₙ ₌ ₁ (bₙ) ¹/ⁿ does not converge.
To determine the exact limit, we would need more information about the sequence (bn). If we can affirm that the nth root of bn is always less than 1, then the series converges. However, without further information about the behavior of bn, a definitive conclusion about convergence cannot be reached.