Limit Comparison Test and Direct Comparison Test

[MrJbackhouse]

I LOVE YOU!!!!!! I wish you were my calculus teacher

[jennygallery]

please make a video about solving improper integrals with the comparison test!!

[patrickJMT]

@jennygallery ok, that sounds like a good idea

[SpnFan04]

is there a video on solving improper integrals using the comparison test?

[patrickJMT]

@SpnFan04 that is a good question

[youngJib20]

haha he says "a-subbinz" and "beesubinz"

[Billabong024]

Just learned this today and ONCE AGAIN, you saved my life!

[dgboygx]

Brilliant. Thanks.

[caligal2010]

Hey Patrick, my test is next week but I was wondering, whats the use of doing the limit of (a sub n) / (b sub n) if you end up only looking at what (b sub n) does in the end? Since (b sub n) converges in your example, why do we need to do all that other stuff? Also, what does the 2 mean? In my problem, I got that the limit when I divide mine is 1. I don't know what that means, lol. SO CONFUSED.

[ferasteddy]

Patrick, this doesn't complement what you said about the Test for divergence, for the second example you said the limit = 2, and it converges. But the Test for Divergence states that if the limit is not = 0 then it diverges, our limit was not = 0 so how come it converges?

[patrickJMT]

@ferasteddy this is not the test for divergence that i am using.

[MrQuotidian]

In the first problem, say the series had been 999/(n^3+4), and you were comparing it to 1/(n^3) over the interval 1 to infinity. At n=1, the first series is 999/(1^3+4) = 999/5 which is greater than 1/(1^3). But if you factor out the 999, that all the sudden makes it valid?

[1Man2Go]

i love you

[abthurd]

okay, I'm just confused.

When can I use either the direct comparison test or the limit comparison test?

[patrickJMT]

@abthurd it depends on the problem

[xSnapper26x]

@abthurd You can use it to determine the convergence or divergence of a series. This is done by comparing it to a series that is known (convergence / divergence) to determine whether it is greater of less than. Depending on the state of that you can determine the convergence or divergence of the series.

[Zirnon]

Your sigmas are cool looking! Great vid!

[tshek129]

these videos are great! thank you so much, you have saved my life!

[ThePeterEffect]

Thank you! Your videos have been helping me since high school! I always recommend your videos to everyone who's having problems with calc.

[0797510396]

Thank you very very much , now i get it

[t0m0wnzj00]

One of your best videos! Thanks!

[stephenxluo]

wouldn't the integral test also work in your first example?

[88899Mo]

you are the best

I really understand that lesson after alot of searches

[wellitstrue102]

patrick, you're my new hero.

[ThinkFreely25]

Alright, in this video...

I love these videos!

[Nandha99]

what if the powers are different on the top and bottom? and also what if they question was for 1/n!, then what would you compare it to?

[VersatilityInfinite]

the limit comparison goes both ways as well right? if the limit as n approaches infinity is > 0 and if b_n diverges does a_n diverge as well?

[akhsnow]

@VersatilityInfinite no, but if in that situation a_n diverges, then b_n diverges. I like to think of it as b_n can force a_n to zero, but if b_n goes off to infinity, a_n could still converge below it. But if a_n diverges, then it forces b_n to also diverge. This is of course only true if b_n is greater than a_n. Not sure if this makes sense but it is the way I remember it.

[l00seg00tyh0les]

Greetings from air academy calc bc! I hate math, but I love your videos!

[JaktheAtheist]

I haven't watched your other videos but you have the best series/sequence videos on youtube. Thanks so much for the hard work.

[irwatchvids]

ohhh thank you so much, you are so good at explaining things!

[The12thAkatsuki]

In the limit comparison test, if the limit is a positive finite number, then both do the same thing. But what if the limit is negative, zero or infinity?

[floyd617]

hey patrick.. I dont understand.. why do we ever have to use a limit test.. i know how you said that to use the direct comparison test you need to see if a_n is smaller or bigger than b_n. but does that really matter because if b_n converges a_n has to converge and if it diverges a_n should also diverge, right? because by doing the limit test we will end up finding if both of them diverge or converge..so my question is do we really have to use the limit test?

[4yipchoclover]

would this be the same thing when dealing with improper integrals? please and thank you :)

[n8ivebl00d]

When you're comparing, do you compare using interval [1, inf), or is it for all n's. If you have to use the interval, why can't you say that 1/n^3 is less than or equal to 1/n , which is divergent?

[lyricnme]

Just wanted to let you know that you have helped me understand in 10 minutes what my professor couldn't teach me in a week. And Patrick, you have no idea how popular you are. I'm in engineering and all of my classmates use your videos to get them through courses.

[bondseatsroids4lunch]

thank you helped out big time... makes sense when my calc bc teacher doesn't

[rickusom1]

dude, has anyone ever told you that you are friggin' awesome for doing this on youtube? thanks

[FriedRicee]

i have my math final in a few hours along with my chem final. you saved my life, thank you

[FriedRicee]

im a lefty too! :D

[Soulscarz]

anyone know what you would pick for you other term if it is (1/ln x)?

[floyd617]

@Soulscarz I would say 1/n no?

[pickyjane]

when do u use which?

[arunderoyal]

learnt a lot from You!!! U Rock...........

[pickyjane]

when do you know when to use which test??

[NeoXC]

So, whenever we pick a b_n, would the b_n always need to be greater than your a_n? I'm a tad confused as to how you'd pick your b_n.

[anjicedtea]

i went through your entire sequence/series playlist. LOVE YOU AND THANKS!

[patrickJMT]

@anjicedtea you are welcome! i think there are a few videos left out of that playlist though!

[amrty1]

@patrickJMT hwy patrick God bless you patrick i have learn all my maths from you but i have problem that i am in BSc Doubble maths and physics nd bad luck is that in my city there is no teacher of maths B for twetion and my professor is not the person like you nd some time it is very hard to understand him, all your videos are from my maths A. so if you are good in maths b Such as MECHANICS so plz tell me and if you are not the tell me where can i find a teacher like you REGARDS

[FaiththeHairstylist]

I like your logic about using the limit comparison test over the direct comparison test. It seems the tricky part is finding the limit to compare it to.

[patrickJMT]

@FaiththeHairstylist typically it seems like most of the time it has something to do with a p-series!

[mdman422]

Patrick

I would just like to say that by far this is the most helpful learning tool that I have come across in all my years in school. You are very concise and explain all your steps you should definitely consider being a teacher I honestly think you would make an excellent one

[91jbarra]

Excellent...crisp clear

[mnava19]

thank you very much patrick , you truly have helped me in my calc 2 class. i dont know what id do without you.

[gsowx2]

If the series was ln n/n^3, would you still compare it to 1/n^3?

[patrickJMT]

@gsowx2 i would do direct comparison test with 1/n^2

[cookiemonsteronthego]

@patrickJMT why?

[ThatsSoBS]

THANK YOU!

[bluegrapes2010]

thank you!

[bluegrapes2010]

you are so awesome Patrick!

[rob5654]

this is awesome, especially since i'm taking and 8 week calc 2 course hahaha

[ammatzner]

Your videos BLOW MY MIND! You explain in 10 minutes what was NOT explained in hours and hours of class, or by painstalkingly self teaching from Thomas Calculus. THANK YOU!!!!

[RylyC]

@ammatzner Thomas' painstaking? It could be worse, you could have Stewart haha. I taught myself using Thomas and loved it. Reviewing from it right now.

[orion2020]

THANK U THANK U THANK U,!!! really helpful!!!!!

[CarbonFiberX]

Patrick, Thanks for your videos on Calculus. I am taking Calc 2 right now in the spring (7 weeks - Not the best idea). Your videos helped out a lot trying to understand the different tests. One of the issues I have is testing for absolute convergence for trig. I have looked at your other videos. An example I came across is the summation of cos ((kπ)/k) going from k=1 to infinity. This I realize is an alternating series. What would you do to test for convergence?

Carbon Fiber X

[CarbonFiberX]

I did the ratio test on the absolute value of cos((kπ)/2) and I found that it Diverges. I then tested it for Conditional Convergence and found that if you take the limit of cos((kπ)/2) you get either 1 over infinity or -1 over infinity which both are equal to one. I then looked at whether it decreases or not. I rewrote the equation as (-1)^(k + 1) times (1/k). I found the numerator of cos kπ just determines whether the sum is positive or negative. I then did the integral test.

[CarbonFiberX]

Oh wait, nevermind. Doing the integral test was bad. I did the comparison test and compared 1/(k+1) to 1/k and since 1/k decreases and since 1/k is larger the smaller function must also decrease. Therefore the test for conditional convergence proves it is conditionally convergent. Sorry for the long posts. Is there an easier way to do do this problem?

[thetruenumberone]

l'hopital!!

[Tasc71]

awesome videos, super helpful!!!

ive got a question though, is there any easy way to tell which test you use when you come up on a series?

ive got a midterm on friday, and im pretty sure my teacher is just going to put tons of series on it. i can sometimes tell which test to use, but othertimes its harder...any suggestions?

[purohiti]

hey patrick! thank you for your vids...i don't know what i would do without them.

but i have one question..could you always use dct or lct? or does it matter depending the problem?

[patrickJMT]

@purohiti yes, it for sure matters on the problem. sometimes, u can use both, sometimes only one seems to be applicable.

[ldc62]

@patrickJMT  how can u tell which one is a better method to use or when u should use D or L???

[the70stang]

I just learned 3 weeks of my Calc 151 class in about 30 minutes thanks to your videos! You are a great teacher and very helpful. Thank you!

[strucklovegeek]

how do you know when to do the direct comparison test and when to do do the limit comparison test?

[260191894]

Why can't professors be so simple? Introductionary courses should be practical, not strongly theoretical. I really enjoy these videos.

[MetalImpulses]

Thanks! You are my God of clarification!

[kigomakid]

Thank you so much... its amazing how clear you make this stuff for me.

[DrBobbyRay]

Life saver Hahaha My Cal III teacher talks way too fast for me. Thanks alot mang

[kungman190]

too fast

[kimpoi00]

as n approaches my limits!! damn!! *confused.. huhu

[crazycool1128]

for limit comparison test, how do we choose Bn??

[HamsAlamal]

we just take the highest power in the numerator

and divided by the highest power denominator

from the original

[kimpoi00]

yea.. how do we choose the Bn?!!?!

[i3loodhunte31181]

highest power on top minus highest power on bottom. ex) n^3/n^2 the bn will be 1/n

[jamescboyd]

In your playlist, you have the order of your DCT/LCT videos mixed up. This one should come first.

Appreciate the help!

[MundusLitterae]

Agreed. I was lost when you started with the other vid. Thanks

[TheLemon22]

if i ace my calc midterm you are definitely getting a donation

[HiShBL2]

Thanx man , you're awesome .

[wad572]

i dont understand why did we choose 1/n3 in the beginning

[jedijack01]

its just sought of intuitively understanding that the series there is going to be less than (1/n^3), there is no real rule for working out what to compare it to... just something that fits and works.. do lots of practice questions, the key is reaching that intuitive understanding phase

[cesfigas]

Your making a public service here, you should be rewarded. This is a public service that helps students from all over the world to achieve success. Greetings from Portugal.

[patrickJMT]

not really.... although my preference is to use the limit comparison, then direct, then integral test; i think LC is easier to use than DC which is easier than IT

[bighorn2004]

these videos saved me so much time! i get it now. you rule.

[ikkayodic]

you have absolutely no idea how good your videos are jk. my teacher used a few of your videos for class =) are you a professor?

[patrickJMT]

i teach part time, yes

[flipponzin24]

Very helpful video =:-) Thank you!

[sidney765]

THANK YOU MAN FOR THESE VIDEOS!!! you saved my life!

[JoeyyT20]

i had the same question as Attackgoat2, why do you have to compare it to something similar? or do you have to?

[patrickJMT]

you can compare it to anything you wish! so long as you know it converges/diverges, and you get a 'useful' value from the limit

[Attackgoat2]

And then attempt the direct comparison again that is?

[Attackgoat2]

Hey Patrick, your videos are a lifesaver. I do have one question though, after you tried to use 1/n^3 as a direct comparison and it didn't work after you changed the sequence to 1/n^3-4, why couldn't you just change your b sub n to 1/n^2 or something that we also know converges by the p test?

[patrickJMT]

try it out and see if it works! i would have to rewatch the video and look at the problem.

there are no hard and fast rules about what to compare a particular series with

[itzxclusive]

stupid question...is this the same technique you would use if the question was an integral instead of a summation?

[patrickJMT]

yes, u could use the same idea to justify that the value of one integral is larger/smaller than another

[AllahinakadarAdanali]

EXCELLENT, EXCELLENT, EXCELLENT... Thanks

[thzel]

ohh finite!lool sorry, got it know!

[thzel]

soory, not english, what do you call number 2 at 5:36? what does that mean?

Thanks great videos!

[1950slollipop]

how do you know when to use the direct comparison test and the limit comparison test?

[patrickJMT]

well, sometimes you can use both. you just have to be able to justify the inequality in the direct comparison test.

in general, if i think a problem can be done both ways, i usually opt for the limit comparison test (but i am also very comfortable at finding limits, which is something that gives some people problems!)

[charcape88]

thank you for making these videos!!

[patrickJMT]

no prob

[The4thbassguy]

does your bsubn have to be greater than your asubn? or can you compare your series with Any other series, as long as you know if its conv or div?

[patrickJMT]

the latter is correct

[LaTaye23]

what if you got a -2 inside of the pos what would the answer be

[karlosgaona]

most helpful and great videos in youtube ! where can I donate ?

[patrickJMT]

actually, the best way to donate is to tell a couple of people about how helpful my vids are : ) that would be the best donation for me!! (and this way, you get to keep your $)

[terminaterjohn]

videos are awesome!!!!!!!!!!!!!

[patrickJMT]

thanks!

[yang2o9]

I like your videos :D Keep on making some more :D

[patrickJMT]

i will! please give them a good rating too! : )

[paisleygnaps]

Thanks for your encouraging words! I appreciated them. =)

[paisleygnaps]

I find your videos very relaxing =) (not to mention enlightening!) Also, as a trainee maths teacher here in Australia, very useful as a nice style of teaching. Thanks heaps!

[patrickJMT]

wonderful! glad the videos help.

and nothing makes me feel better than to know that i am helping out another math teacher!

good luck!! teaching is the most noble profession! : )

[AznHavx90]

You teach better then my professor!!!

[NaturallySequential]

Thank you.

[patrickJMT]

no problem!

[AyvenActually]

good videos; these are very helpful.

[wu7]

great review, thanks

[brvheavyhitter20]

thanks for the vids...i have a test on Sequences and Series tomm in Cal 2 and ur videos really help!! keep up the good work