Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?
$ a_n = n(-1)^n $
the sequence is not bounded.
for this sequence will actually show that it's not monotone and that it's also not bounded now. The way to show that it's not monotone is to show that it's not increasing and decreasing, so a one equals one. Excuse me, a negative one. A two is equal to two, so that's an increase. However, a three equals minus three, and that's a decrease. So the sequence is not decreasing because at some point there's the increase. On the other hand, the sequence is not increasing because there's a decrease. So therefore, a N is neither increasing or decreasing, so it's not monotone now. On the other hand, let's show that it's not bounded. So instead of looking at all the ends, let's just look at the ends of the form to. And so, for example, I mean, like a two, a four, a six and so on notice that these are all just equal to two floor six and so on, and therefore the limit of a to end is just equal to the limit of to end. These air limits is and goes to infinity and the limit of two, and it's just infinity, so there's no way that the sequence could be bounded because we have this. Some of the terms are getting larger and larger, closer to infinity. So a N is not bounded, and that's our final answer, not bounded in that monotone.