Homework SourseDoes the sequence ecosn have a convergent subsequence Explai
Does the sequence {e^cos(n)} have a convergent subsequence? Explain your answer.![Does the sequence {e^cos(n)} have a convergent subsequence? Explain your answer.SolutionYes, sin cos(n) is dense in the interval [-1,1], therefore, {e^(cos(n))](/WebImages/28/does-the-sequence-ecosn-have-a-convergent-subsequence-explai-1075537-1761564039-0.webp "Does the sequence {e^cos(n)} have a convergent subsequence? Explain your answer.SolutionYes, sin cos(n) is dense in the interval [-1,1], therefore, {e^(cos(n))")
Solution
Yes, sin cos(n) is dense in the interval [-1,1], therefore, {e^(cos(n))} have indeed subsequences which are convergent. It can also be explained as we know that periodic functions (and functions composed of periodic functions) have convergent subsequences on unit circle. Therefore, the given sequence {e^cos(n)} will have a convergent subsequence.
