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JH
Numerade Educator

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Problem 72 Medium Difficulty

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = \cos n $

Answer

Bounded but not monotonic

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Video Transcript

first, let's show that this sequence is not monotone IQ. So a one that's co sign one. And that's approximately point five four o three Now a two. This time you just do co sign of two. So let's go out and plug that in and then round off to a calculator. Negative point four one six one a three that's co signed three. And once again, back to the calculator negative point nine eight nine nine and then a four co sign. For now, let's go ahead and approximate that. Using the calculator once again, we're almost and that this will be enough for us to make to stop now. Originally, we could see that it was decreasing, and then it even decreased again, so it looks like it had a chance. But then it switched Teo increasing. Therefore, it cannot be decreasing because it increases. On the other hand, it can be increasing because it decreases. They're for a end is not monitor Nick. Now for the second question, we know that a N is less than or equal to one. An absolute value, says co sign is less than or equal to one and absolute value. So this sequence is bounded because we just showed that it's bounded by one. So it's not monotone yet. It's bounded. That's our final answer.