Question

determine whether the sequence converges or diverges. if it converges, find the limit. (if an answer does not exist, enter dne.) an = cos2(n) 5n lim n→[infinity] an =

Answer 1

The sequence an = cos2(n) / 5n converges to 0 as n approaches infinity.

Step-by-step explanation:

We need to determine whether the sequence an = cos2(n) / 5n converges or diverges as n approaches infinity. The numerator, cos2(n), oscillates between 0 and 1, while the denominator, 5n, grows without bound. As n becomes very large, the denominator will dominate the fraction, driving the value of an closer and closer to zero. This means that the sequence an converges to 0 as n approaches infinity. Hence, the limit of an as n goes to infinity is 0.

Answer 2

The sequence with term a_n = cos^2(n) 5^n diverges as n approaches infinity because the factor 5^n grows without bound.

Step-by-step explanation:

The sequence an = cos^2(n) * 5^n is analyzed to determine its convergence or divergence as n approaches infinity. As n increases, the factor 5^n grows without bound, while cos^2(n) oscillates between 0 and 1. The dominance of 5^n causes the terms to become increasingly larger, leading to divergence. The behavior of the terms is characterized by the unbounded growth of the exponential factor, overshadowing the oscillations of the cosine term. Consequently, as n approaches infinity, the sequence diverges due to the overwhelming influence of the exponential growth.