**Session Date:**4/26/2016 5:49PM ET

**Length:**44.8 minute(s)

**Subject:**Math - Calculus

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Guest (Customer)

[00:00:00] Hi, I need help with these problems. [ File > http://lhh.tutor.com/SharedSessionFiles/6c98f7ec-4419-4df4-a6e7-afa0ba96a812_Screen_Shot_2016-04-26_at_5.22.07_PM.png ]

Lucia F (Tutor)

[00:00:12] Welcome!
Let's get started.

[00:00:19] Give me a minute while I look at your problem

Guest (Customer)

[00:00:34] hi! okay no problem

Lucia F (Tutor)

[00:03:00] I'm working on this...

[00:03:08] Alright

[00:03:13] Do you know how to start this problem?

Guest (Customer)

[00:03:20] i started the first one but i am stuck

[00:03:38] i'll show you how far i got

Lucia F (Tutor)

[00:04:01] Okay!

[00:04:10] Oh sorry, added the image in the middle of your writing!

Guest (Customer)

[00:04:47] now i am not sure what to do

Lucia F (Tutor)

[00:05:25] What method did you use to try and find the interval of convergence? What does that formula account for?

Guest (Customer)

[00:05:41] i used the ratio test

[00:05:52] i am stuck i have no idea how to do this

[00:05:59] like how to find the interval of convergence

Lucia F (Tutor)

[00:06:09] Okay, let's work on it togheter

[00:06:40] So you have to find the interval of conevrgence and tried the radio test... Let me check the formula you got is right

Guest (Customer)

[00:06:45] ok

Lucia F (Tutor)

[00:08:11] Can I ask why you took the absolute value of x?

Guest (Customer)

[00:08:38] because the ratio test is the lim as n approaches infinity of the absolute value of an+1/an

Lucia F (Tutor)

[00:09:11] Oh, you're right I forgot that one detail of the absolute value in the ratio test

[00:09:23] Ok, so you got the limit in the ratio test is the absolute value of x

[00:09:38] and what does the ratio test tell you about convergence?

Guest (Customer)

[00:09:54] i don't know, i'm confused

Lucia F (Tutor)

[00:10:45] That's alright Do you know how to use the ratio test? When does it say the series converges?

Guest (Customer)

[00:11:00] no idea haha

Lucia F (Tutor)

[00:11:26] Alright, let's start from there!

[00:11:33] Like you said, the radio test is this...

[00:12:15] You know that definition, right?

Guest (Customer)

[00:12:28] yes

Lucia F (Tutor)

[00:12:41] When L < 1, the infinite sum for an convernges

[00:12:54] when L > 1, the infinite sum for a_n diverges

[00:12:56] correct?

Guest (Customer)

[00:13:06] yes

Lucia F (Tutor)

[00:13:34] And you got |x| for the series in your problem

[00:13:48] How can you use that to find out where is diverges and where it converges?

Guest (Customer)

[00:14:29] i don't know

Lucia F (Tutor)

[00:15:10] I'm sure you can do it Tell me again, when you use the ratio test, you know the series converges when you get what kind of number?

Guest (Customer)

[00:15:23] one that is less than 1

Lucia F (Tutor)

[00:15:42] Good! And for the series you want to test, what did you get?

Guest (Customer)

[00:15:49] x

Lucia F (Tutor)

[00:16:18] Great! so if you wanted the series to converge, what would x have to be?

[00:16:22] |x|*

Guest (Customer)

[00:16:32] less than 1?

Lucia F (Tutor)

[00:16:39] Excellent!

[00:17:06] Do you see now the condition you got for x?

Guest (Customer)

[00:17:11] yes

[00:17:16] so what is the interval?

Lucia F (Tutor)

[00:17:19] That's a first step towards the inteval of convergence

[00:17:38] well, if it has to be |x| < 1, what would x be between?

Guest (Customer)

[00:18:02] from -1 to 1

Lucia F (Tutor)

[00:18:07] Good job!

Guest (Customer)

[00:18:18] but how do we know whether each of those values is included or excluded

[00:18:28] like when we write it, if it has a parentheses or bracket

Lucia F (Tutor)

[00:19:04] What has to have parentheses or brackets? I don't understand, sorry

Guest (Customer)

[00:19:20] if it's (-1, 1] or (-1,1) or [-1,1] etc

Lucia F (Tutor)

[00:19:27] Oh!

[00:20:03] Well, you know from the ratio test that if L is exactly 1 then the test is not useful, right? L has to be exactly less than 1

[00:20:16] so if |x| has to be exactly less than one...

[00:20:34] what would that mean, say for positive x's?

Guest (Customer)

[00:20:58] they have to be less than 1?

Lucia F (Tutor)

[00:21:24] That's right, does that mean they can be 1?

Guest (Customer)

[00:21:36] no

Lucia F (Tutor)

[00:21:56] Great! So would you use parenthesis or brackets for 1 in the interval?

Guest (Customer)

[00:22:11] which one means what?

Lucia F (Tutor)

[00:22:38] Parenthesis means it's not part of the interval, bracket means it is

Guest (Customer)

[00:22:47] so then it will be parenthesis

Lucia F (Tutor)

[00:22:54] Great job!

Guest (Customer)

[00:23:07] so [-1, 1)?

Lucia F (Tutor)

[00:23:39] The -1 is the same. Do you know how to evaluate inequalities iwth absolute values?

[00:23:48] I can show you how

Guest (Customer)

[00:24:27] no

Lucia F (Tutor)

[00:24:59] Alright, let me show you quickly how to do them

[00:25:09] Let's do the same as this example

[00:25:30] You have to divide it in two.. One for positive x's ....

[00:25:56] Which means the absolute value remains the same, since it's already positive

[00:26:06] And one for negative values...

[00:26:38] When you remove the absolute value bars from a negative unknown number (any negative x) you add a minus sign in front of it

[00:26:51] That way, x becomes positive!

Guest (Customer)

[00:26:54] ok

Lucia F (Tutor)

[00:27:25] Now, how do you clear out the x in the negative one?

Guest (Customer)

[00:27:36] divide by negative 1

Lucia F (Tutor)

[00:27:56] Good! and what happens? You have to be careful with that step

Guest (Customer)

[00:28:04] inequality flips

Lucia F (Tutor)

[00:28:09] Excellent!!

[00:28:24] And you get this, is that right?

Guest (Customer)

[00:28:29] yes

Lucia F (Tutor)

[00:28:49] No it's not haahah I wrote the inequality sign wrong, sorry

Guest (Customer)

[00:29:01] oops haha

Lucia F (Tutor)

[00:29:11] Very well

[00:29:18] So what is the resulting interval from this?

Guest (Customer)

[00:29:26] [-1,1]?

Lucia F (Tutor)

[00:29:47] That's almost right! Remember they are not included!

Guest (Customer)

[00:29:56] (-1,1)?

Lucia F (Tutor)

[00:30:03] If they were included, the inequality sign would be this:

[00:30:11] Correct!

Guest (Customer)

[00:30:20] yay

Lucia F (Tutor)

[00:30:33] those I wrote on the board, with an extra line, mean less than or equal to, and greater than or equal two

Guest (Customer)

[00:30:38] yes

Lucia F (Tutor)

[00:30:39] they are different

[00:30:44] Great!

[00:30:59] Only when they can be equal to that you use brackets

[00:31:14] Alright then, want to try the next problem and see if you got the concepts right?

Guest (Customer)

[00:31:29] yes i actually tried it already and wanted to check it with you

Lucia F (Tutor)

[00:31:42] Good! Let's see then

Guest (Customer)

[00:32:12] i'm going to show you the final steps because the earlier ones are all just rearrangement

Lucia F (Tutor)

[00:32:39] Okay!

Guest (Customer)

[00:33:44] tell me if that's right

Lucia F (Tutor)

[00:34:22] The answer you got is correct! But remember to not omit terms when writing the limits since teacher don't normally like that in exams

Guest (Customer)

[00:34:37] like which terms?

Lucia F (Tutor)

[00:34:59] Here, you didn't keep writing the |x|

Guest (Customer)

[00:35:03] right

Lucia F (Tutor)

[00:35:17] Even though it's right the limit of the following part is 1

[00:35:24] But What you get is |x| as the result

[00:35:32] and that is what has to be less than one

[00:35:51] It's a common thing students don't think they have to write and teachers don't like it

[00:36:16] Alright, so did you understand everything we did to get the interval of convergence?

[00:36:32] Can you write down the final result for that one as an interval too?

Guest (Customer)

[00:37:46] (-1,1)?

Lucia F (Tutor)

[00:37:52] Great!!

[00:37:58] You did really well!

Guest (Customer)

[00:38:11] thank you

Lucia F (Tutor)

[00:38:35] Was everything we did clear to you? Do you need help with anything else?

Guest (Customer)

[00:40:32] yes i need help with one more thing that i am stuck with relating to this

[00:40:46] it's a different problem but i am having trouble with the final inequality that i got

Lucia F (Tutor)

[00:41:06] Okay, let's see it!

Guest (Customer)

[00:42:09] is that right??

[00:42:24] because one of my friends got (-1,5) instead of (-5, 5)

Lucia F (Tutor)

[00:42:31] The one on the left yes, it's right

[00:42:42] Your friend is right

[00:42:45] Let's see why

[00:43:18] Like I said before, when you consider the expression inside the absolute value is negative, you have to add a minus in front of it

[00:43:23] that would be like this:

[00:43:54] Can you spot where you got it wrong in your expression?

Guest (Customer)

[00:44:20] yes i understand now

[00:44:42] thank you!!!

Lucia F (Tutor)

[00:44:47] Excelent!

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