Final answer:
The sequence a₍n₎=cos²n/2ⁿ converges because the numerator is bounded and the denominator grows without bound, forcing the terms to approach 0 as n approaches infinity. The limit of the sequence is 0.
Step-by-step explanation:
The sequence given is a₍n₎=cos²n/2ⁿ. To determine convergence or divergence of the sequence, we need to analyze the behavior of the terms as n approaches infinity. The numerator cos²n oscillates between 0 and 1, but due to the square, it stays positive. Since the cosine function is bounded, and the denominator 2ⁿ grows without bound, we can infer that the overall fraction is getting smaller as n increases. We can see that every term in this sequence is bounded above by 1/2ⁿ. As n goes to infinity, 1/2ⁿ goes to 0. Therefore, the limit of the sequence a₍n₎ as n approaches infinity is 0. Thus, the sequence converges, and the limit is 0.