WEBVTT
1
00:00:00.140 --> 00:00:03.830 A:middle L:90%
for this sequence will actually show that it's not monotone
2
00:00:03.830 --> 00:00:07.309 A:middle L:90%
and that it's also not bounded now. The way
3
00:00:07.309 --> 00:00:11.359 A:middle L:90%
to show that it's not monotone is to show that
4
00:00:11.359 --> 00:00:16.019 A:middle L:90%
it's not increasing and decreasing, so a one equals
5
00:00:16.019 --> 00:00:19.309 A:middle L:90%
one. Excuse me, a negative one. A
6
00:00:19.309 --> 00:00:22.250 A:middle L:90%
two is equal to two, so that's an increase
7
00:00:25.289 --> 00:00:30.489 A:middle L:90%
. However, a three equals minus three, and
8
00:00:30.489 --> 00:00:35.409 A:middle L:90%
that's a decrease. So the sequence is not decreasing
9
00:00:35.409 --> 00:00:38.700 A:middle L:90%
because at some point there's the increase. On the
10
00:00:38.700 --> 00:00:41.729 A:middle L:90%
other hand, the sequence is not increasing because there's
11
00:00:41.729 --> 00:00:48.659 A:middle L:90%
a decrease. So therefore, a N is neither
12
00:00:48.670 --> 00:01:00.200 A:middle L:90%
increasing or decreasing, so it's not monotone now.
13
00:01:00.200 --> 00:01:02.990 A:middle L:90%
On the other hand, let's show that it's not
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00:01:02.990 --> 00:01:07.000 A:middle L:90%
bounded. So instead of looking at all the ends
15
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, let's just look at the ends of the form
16
00:01:08.590 --> 00:01:11.700 A:middle L:90%
to. And so, for example, I mean
17
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, like a two, a four, a six
18
00:01:15.260 --> 00:01:19.549 A:middle L:90%
and so on notice that these are all just equal
19
00:01:19.549 --> 00:01:26.989 A:middle L:90%
to two floor six and so on, and therefore
20
00:01:26.989 --> 00:01:30.590 A:middle L:90%
the limit of a to end is just equal to
21
00:01:30.590 --> 00:01:34.560 A:middle L:90%
the limit of to end. These air limits is
22
00:01:34.560 --> 00:01:38.790 A:middle L:90%
and goes to infinity and the limit of two,
23
00:01:38.790 --> 00:01:42.540 A:middle L:90%
and it's just infinity, so there's no way that
24
00:01:42.540 --> 00:01:45.180 A:middle L:90%
the sequence could be bounded because we have this.
25
00:01:45.390 --> 00:01:48.650 A:middle L:90%
Some of the terms are getting larger and larger,
26
00:01:48.650 --> 00:01:55.409 A:middle L:90%
closer to infinity. So a N is not bounded
27
00:01:59.680 --> 00:02:01.780 A:middle L:90%
, and that's our final answer, not bounded in
28
00:02:01.780 --> 00:02:02.560 A:middle L:90%
that monotone.