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System Message

[00:00:00] *** Please note: All sessions are recorded for quality control. ***

Guest (Customer)

[00:00:00] Does this converge or diverge?

William R (Tutor)

[00:00:05] Hi there!

Guest (Customer)

[00:00:07] hi

William R (Tutor)

[00:00:11] Welcome to tutor.com .
How are you?

Guest (Customer)

[00:00:13] good

William R (Tutor)

[00:00:31] So I see you're having trouble with this problem, correct?

Guest (Customer)

[00:00:39] yup

William R (Tutor)

[00:00:50] Ok, that's not a problem!

[00:01:08] Did you have an idea of where to begin?

Guest (Customer)

[00:01:19] um.. I'm not really sure which
strategy to use..

William R (Tutor)

[00:01:47] Which is completely understandable.
There are something like a dozen different tests for convergence and divergence to choose from.

[00:02:12] What tests have you learned so far?

Guest (Customer)

[00:02:25] I think we're learned just
about all of them

William R (Tutor)

[00:02:29] That will be the best place to start.

[00:02:42] Ok, so we need to narrow it down.

Guest (Customer)

[00:02:58] well.. i doubt root or ratio
test will work

William R (Tutor)

[00:03:12] Why do you think that is?

Guest (Customer)

[00:03:25] root test only works if its
raised to the power of n

William R (Tutor)

[00:03:31] That's right!

Guest (Customer)

[00:03:39] and its hard to compare
tan(1/n) to tan(1/(n+1))

William R (Tutor)

[00:04:06] It is, yeah.
You can still do it, but as it turns out, taking that limit gives you a final answer of 1.

[00:04:14] Do you know what happens in that case?

Guest (Customer)

[00:04:36] the test is inconclusive

William R (Tutor)

[00:04:45] Exactly, so your initial thought was right.

[00:05:10] So the easiest test to try to apply would probably be the divergence test, which is also sometimes called the nth term test.

Guest (Customer)

[00:05:32] ok

William R (Tutor)

[00:05:37] Are you familiar with that test?

Guest (Customer)

[00:05:43] yes

[00:05:54] if the limit as n -> infinity is
not zero, the series diverge

William R (Tutor)

[00:06:01] Exactly!

[00:06:10] What happens if you were to apply that test here?

Guest (Customer)

[00:06:23] well

[00:06:27] you get tan(0)

[00:06:29] and that equals 0?

William R (Tutor)

[00:06:34] That's right!

[00:06:37] What happens in that case?

Guest (Customer)

[00:06:45] it might converge or diverge

William R (Tutor)

[00:07:05] Exactly, it doesn't tell you conclusively, so we have to use another test.

[00:07:10] Is this series geometric?

Guest (Customer)

[00:07:19] um.. I don't think so

William R (Tutor)

[00:07:25] You're right, yeah.

[00:07:29] So that test doesn't apply.

[00:07:35] What about the integral test?

[00:07:49] Is this something you could integrate easily?

Guest (Customer)

[00:07:52] i don't think so

[00:07:54] because of 1/n inside

William R (Tutor)

[00:08:03] Yeah, exactly.
So that one's out!

[00:08:12] We've at least gotten rid of a lot of tests so far.

[00:08:38] That leaves us with p-series, comparison, limit comparison, and alternating series/absolute convergence tests.

[00:08:52] Is this a p-series?

Guest (Customer)

[00:08:55] nope

[00:09:05] and this isn't an alternating
series

William R (Tutor)

[00:09:17] Exactly, leaving us only with the comparison and limit comparison tests.

[00:09:27] So now we have to figure out what to compare this series to.

[00:09:33] Do you have any ideas?

Guest (Customer)

[00:09:52] maybe tan(n)?

William R (Tutor)

[00:10:16] I actually wouldn't use tan(n), and here's why.

[00:10:42] As we plug in different values of n for tan(n), eventually we start to get negative values as well as positive ones.

[00:10:57] That doesn't happen with tan(1/n), because the inside continues to get smaller instead of larger.

Guest (Customer)

[00:11:18] ok

William R (Tutor)

[00:11:24] Both of the comparison tests require the series we work with to have positive terms, meaning tan(n) is out unfortunately.

[00:11:33] Other ideas?

Guest (Customer)

[00:12:14] hm...

William R (Tutor)

[00:12:31] We just tried working with the tan part of it and determined that didn't lead us anywhere.

[00:12:45] So how about trying to use the 1/n part instead?

[00:13:24] This might work since the terms of the series of 1/n are all positive at least.

[00:13:33] What happens to the series of 1/n?

Guest (Customer)

[00:14:13] it diverges

William R (Tutor)

[00:14:17] That's right!

[00:14:50] Now, to use the comparison test, what would we need to know about the original series?

Guest (Customer)

[00:15:21] if it's larger or smaller than
the other one?

William R (Tutor)

[00:15:26] Exactly.

[00:15:33] That's not very easy to do here, is it?

Guest (Customer)

[00:16:17] nope

William R (Tutor)

[00:16:36] Since it's not very easy to do, that really only leaves us with one choice: the limit comparison test.

[00:16:43] What do we need to check for that test?

Guest (Customer)

[00:16:59] if the two limits, when
divided, equal a constant

William R (Tutor)

[00:17:14] Exactly.
Can you try that?

Guest (Customer)

[00:17:59] i'm
not sure what exactly I
should compare it to

William R (Tutor)

[00:18:18] Well, we're comparing it to the series of 1/n.
We have both series that we're going to work with.

[00:18:34] So we just need to evaluate the limit using those two.

Guest (Customer)

[00:18:38] ok

[00:18:42] so it'll be ntan(1/n)

William R (Tutor)

[00:18:55] It would, but I would write it like this:

[00:19:20] If you were to evaluate the limit in that form, what would you get?

Guest (Customer)

[00:19:55] 0

William R (Tutor)

[00:20:09] That's close, but it's off by a bit.

[00:20:17] Earlier, you said that tan(1/n) would go to 0, right?

Guest (Customer)

[00:20:21] right

William R (Tutor)

[00:20:40] What about the denominator?

[00:20:43] What would that go to?

Guest (Customer)

[00:20:45] 0

[00:20:48] so it would be undetermiend?

William R (Tutor)

[00:20:55] It is indeterminate, yep.

[00:21:01] Do you know what we could use now?

Guest (Customer)

[00:21:11] l'hopital's rule?

William R (Tutor)

[00:21:15] Exactly!

[00:21:26] Can you show me what we would get when we apply that rule?

Guest (Customer)

[00:22:18] sec^2(1/n)

William R (Tutor)

[00:22:27] Exactly!

[00:22:34] What does that go to as n approaches infinity?

Guest (Customer)

[00:22:52] 1

William R (Tutor)

[00:23:08] That's exactly right.

[00:23:20] So what does the limit comparison test tell you about the original series?

Guest (Customer)

[00:23:24] it diverges

William R (Tutor)

[00:23:34] Exactly right!

[00:23:49] So does that clear up your question?

Guest (Customer)

[00:24:04] yeah

William R (Tutor)

[00:24:11] Great.

[00:24:22] You have a really good handle on this stuff.

[00:24:35] When in doubt, just go through the various tests you have and try to narrow it down.

Guest (Customer)

[00:24:43] ok got it

William R (Tutor)

[00:24:47] Great!

[00:24:57] I hope you have a great evening.

[00:25:01] Thanks for using tutor.com !