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

$ a_n = n^3 - 3n + 3 $

Monotonic but not bounded

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Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

first, let's determine whether it's increasing or decreasing. So here, what we can do is just replace this with the function f of X By replacing all the ends with ex. Now we have f prime of X equals three X squared minus three, and this is bigger than her equal to zero. If X is bigger than or equal to one, so F is increasing. Therefore, a N is increasing there and which implies that it's monotonic. So that's the first part of this question increasing. And it is monotonic. The next question is, is the sequence bounded? Well, notice that the limit as n goes to infinity of a n equals the limit. X goes to infinity. F of X equals infinity. So this means that a N is not bounded since it's not bounded above, so not bounded. But it is monotone increasing, and that's our final answer