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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

$ a_n = 2 + \frac{(-1)^n}{n} $

Bounded but not monotonic

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Missouri State University

Harvey Mudd College

University of Nottingham

Let's consider the sequence given by N now. The first part is determine whether this thing is increasing, decreasing. Or perhaps it's not even Mon Atomic. Well, let's just write out a few terms here. A one that's just two minus one equals one. Now, how about a two? This one will be to plus a half. So five halfs. So from the first term, to the seconds her we saw increase, how about from the second terms of the third term? This time when and his three will have to minus the thirds, This is five thirds, and this is a decrease. So because of the term from a one to a two, this showed us that it was not decreasing because it increase. And from a two two a three, we see that it's not increasing. Therefore, the sequence A M is neither increasing nor decreasing, so it's not monitor tonic. Now let's go to the next part. We see that we start off at two, and then we're adding a term that goes to zero. So for this problem, it might be best to just noticed that the limit of an equals two so a N is bounded, and this is just coming from the fact that convergent sequences are bounded. This is an important that and we're using that to show and his bounded because it converges. And that's our final answer, not mon atomic, but it is bounding.