Calculus 2 Lecture 9.4: The Comparison Test for Series and The Limit Comparison Test

[@jasontaylor6815]

Tell me: "Does that make sense to you?" I LOVE this method of teaching. Thank you.

[@zakusa9891]

not when he says it every second and no one says anything

[@DAT-OFFICIAL]

@@zakusa9891 They could just be nodding behind the camera. They don't have to audibly say yes. Source: am a college student.

[@zakusa9891]

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[@vaiterius]

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[@zakusa9891]

@@vaiterius textbook >>>>> professor Leonard

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you don't really know how many people you help. I have been watching you since my junior year of high school and never once was I lost. I recommend you to everyone who is lost and struggling. Professor Leonard you are a real one, pls come teach calc 3 at MSU.

[@ProfessorLeonard]

Thanks! I appreciate the comment and the recommendations :)

[@teachingkidsyoga]

IDK where you learned to teach, but you make Calc 2 seem so easy now, i had a hard time understanding until I watched your videos. Thank you.

[@adolphmatjukotja3093]

same here babez

[@ducksoop.x]

Same, I have a feeling I'll make it now. Did horrible on my first exam.

[@Peter_1986]

+Gem I like how Leonard always gives a bunch of examples for everything, I really think that this is absolutely crucial for any scientific subject.

[@yagmurbugucoskun3188]

48:57 He summarises what to do step by step from Lecture 9.1 to 9.4

[@yagmurbugucoskun3188]

Go to the lecture 9.6 1:20:25 for the summary of what geometric, p, harmonic and telescoping series are

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[@MileeWilson]

Probably the best and most sane Math teacher I know.

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9 years later, and still the best Mathematics Professor I could find on the damn website, idk people in high school told me that no one in a college would ever explain things to you with example, well, guess I found the diamond

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Thank you so much for these videos Professor Leonard! Now in 2020 they are still helping students all around the world during this pandemic crisis. God bless you.

[@lllWiseManlll]

Professor Leonard : That's the IDEA Me : level of understanding = 99.99% Professor Leonard: *goes over the example one more time really quick* Does that make sense to you? Me: level of understanding = 100% THANK YOU SO MUCH GTL . Great Teach Leonard!!

[@mmartinez8549]

I usually watch PatrickJMT's shorter videos for calc review, but taking the time to watch your videos has made me more at ease than anything else I've come across for understanding concepts that once seemed so abstract. Thanks Professor!!

[@CattyMcDaddy]

Best mathematics teacher on the planet. "show of hands if you understand this" always slowing down if his students don't get something, which is RARE because the way he teaches is flawless. If I had you as my calc teacher for 1 through 3, I would be a mathematician.Thank you for being amazing

[@lycybin8347]

frick this guy has helped me so much. Wish I could send a letter or something just thanking him. I was struggling so much in calc 2 since it is so much more dense and you need to remember pretty much EVERYTHING from calc 1 and ALL other prerequisites in their most advanced forms and prof Leonard just goes through it so smoothly and breaks it down so well. On my first calc 2 test I received a 48, and then started watching him after that and on my second test I got an 85! Plus his sense of humor makes it even easier to watch and follow along with his lectures and not get bored. Prof. Leonard if you see this, I am so thankful for you, sir!!!!!!

[@meksudbuser9624]

Professor Leonard, you are creating future scientists. You are very organized, clear, and logical. The best mathematics professor I have ever seen in my educational journey.

[@jimmidge13]

I.m 83 and remember failing Calc back in 1951 ..Wish you were there. One knock Ya jus gotta write largerjimmidge

[@billh1337]

what was a calc class like back then? Different from today or just about the same? Math is math?

[@hammam92]

I am wondering why are you here ? i mean why you are watching these lessons !!

[@dominusfons4455]

@@hammam92 I'm guessing they're taking some higher physics class or something or just love math because my pops' 74 and he's taking like quantum chromodynamics and he needed to learn complex calculus and etc to get the prerequisites for the prerequisite class for that class lol

[@hammam92]

@@dominusfons4455 OMG

[@al-zubairalbalushi8034]

@@billh1337 They were studied it in white and black classes

[@foeva2554]

This is literally the first time, after repeating the course 3 times, that i have understood direct comparison test. Thank you so much professor Leonard

[@dantesinaguglia3474]

Such a great Professor, I go to university in Ontario, Canada, (Zoom currently) I don't even attend my calc 2 lectures anymore, he teaches so well! Ive recommended so many classmates.

[@jonnymat1]

Here's a little trick I use to remember when a Direct Comparison Test shows convergence or divergence: Just remember: 1.) " a Smaller Cat" * When a(sub n) is Smaller & b(sub n) Converges: Converges 2.) "a Bigger Dog” * When a(sub n) is Bigger (or larger) & b(sub n) Diverges: Diverges Hope this helped! I used cat and dog because it was easy for me to remember but feel free to use your imagination 

[@freekeyboards]

He does an amazing job at 57:30 of tying all this together based on a student's question. He really weaves the concept of sequences, series, limits, converging, and diverging really well. That part alone is worth its weight in gold.

[@amadif1793]

I can't believe I only started using this guy's videos 2 days before my calc final. The style and pace of teaching is excellent. Thankfully he has a calc 3 playlist, which I will be using a lot!

[@iceverything2000]

limit comparison test 40:55

[@kraven2144]

+iceverything2000 Bless your heart

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FanDroid Thank you very much :)

[@iceverything2000]

Glad I can help :)

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Thank you

[@namehere630]

thankss

[@MoXyiD]

Professor Leonard, I have watched many instructors-- but few have truly done something amazing with their style of teaching and presence in a classroom. You force engagement and respect your students. This alone earns my respect. Thanks for what you do.

[@mrksbrnd91]

Im glad I found these videos 3 weeks before my first exam! I'm getting an A in Calc! #FACTS!

[@neokattan5672]

I like his method of teaching so much because if I'm in the middle of a test and i forget if if something is a + or a - or if something is square root or squared, I know where it all started so I can figure it out. Instead most teachers make you memorize 6 equations when you could be learning this way and only memorizing the 1 that they were derived from and being able to derive them yourself.

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When the teacher sincerely teaches this is the result! You are awesome, Prof!

[@shadabzahra6274]

One semester had passed and I learnt not a single word in university but here everything is sorted. Love you Sir

[@AjaxNotFrancis]

From the bottom of my heart, thank you. The way you present the material and make sure everyone understands is the best way to teach. You make this interesting and I'm actually learning. Again, thank you.

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1:10:03 This right here is what earns you math teacher of a century! Your personality, humor and teaching excellence is what makes you stand apart from all other (math) professors!

[@ZahraZenZest]

You are a savior Professor, specially in this quarantine when most of the professors are just loading us with assignments and are not being helpful, thank you so much. love how you make things easier for us. I Wish you the best in life.

[@4bbiss]

This man is literally the best teacher/educator I've ever had

[@funnypython3190]

Professor Leonard, the least thing i can say that you are a LEGEND,THANK YOU SO MUCH PROFESSOR LEONARD FOR THIS AMAZING GREAT TEACHING OF CALCULUS 1,2,3...., i have been with since precalc and here i am going to differential equations and i want to tell u that i scored the highest grade is all of these calculus courses and this is for a single reason, You are the teacher, thank you Professor

[@blakehudgins9416]

I can not preach this enough - this guy is literally the GOAT calculus professor.

[@enjoycaloy5108]

Around 57:49, the student was confused why 1/((sqrt(n)+1) diverges. if you were confused by it too, the reason why was that the limit of (an/bn) HAS to be a finite number greater than 0 and bn (1/(sqrt(n)))diverges.

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This channel deserves way more views!!! btw, this bro lifts

[@neokattan5672]

that's one of the reasons why most of us pay so much attention to him

[@drhf1214]

i am in calc 2 in UC Davis and Im really thankful for this!

[@drhf1214]

aw man good luck! KZbin professors always have saved my ass. I'm in MAT 22B now and am doing well. Best of luck.

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U graduate yet?

[@SuperMahler2012]

I had MAT 21B and 21C at UCD five years ago, and I seriously wish I found Prof. Leonard during those quarters. Better late than never!

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Where are you rn?

[@emmaburgard980]

1:11:01 a student replied "Damn" and I just felt that in my soul.

[@sunnysamuel7902]

Prof, if your dad is watching these videos as u said, He will be really proud of u to learn math from you !!!!!! You definitely have a wonderful family .. Thank you for all these videos, without which I would have never made Calculus 2

[@georgesadler7830]

Professor Leonard, thank you for an incredible outline and explanation of The Comparison Test for Series and The Limit Comparison Test in Calculus Two. From watching this great video and doing many problems from different Calculus books, I finally have an excellent understanding how to apply these two classical tests for convergence/divergence. This topic is also highly problematic for many students in Calculus Two.

[@dannyphan4922]

I'm not in your course, but, my Calc teacher is has... questionable teaching methods. Thank god for your uploads, it has cleared a lot of things up.

[@coleabrahams9331]

at 1:23:00, for those of you still uncertain about whether sqrt(n) increases at a faster rate than ln(n), take the limit of one of the functions over the other as n approaches infinity. If the limit = 0, the bottom function grows faster. If the limit = infinity, the top function grows faster.

[@w.a612]

I dont get why we cant just plug in high numbers and see that way as a student suggested

[@seabasschukwu6988]

@@w.a612 u can for this prob but it might not always work for u

[@TJ_The_King]

tons of teachers will skip over the past stuff from algebra and cal 1 and expect you to know it but this guy understands that with all the new information it is so easy to overlook the easy stuff these kids are lucky to have someone who will take the time to break down everything.

[@betauser6233]

I dont know which one I want to do first after watching your lessons. Go do my calc homework or go the gym and work them arms!!

[@vanessatram792]

u guys have such a cool lecturer. I would study so hard and always sat at the first row if he was my lecturer. ^^

[@MrChickenstalker]

same!

[@HA-zg4fi]

this was extremely helpful i was so confused. i owe u my first born child now for helping me pass calculus II

[@brookef2011]

I don't know where you teach but I want to transfer there just to take calc II with you. I've tried taking this class so many times. Thanks for recording your classes... it REALLLLLY helps!

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I wake up to your videos every morning. Please keep em coming.

[@flyingpirate8115]

ı really dont know what ı would be doing now without you since I have math exam this weekend... so thank u so much.... I actually learn them.

[@jarodsparacio851]

I'm doing my Calc 2 take-home exam while watching this video. Thank you, Professor Leonard!

[@Jeiss_V]

so...how was your take-home exam

[@jarodsparacio851]

@@Jeiss_V Lol this was 2 years ago, and I passed Calc 2, so I'm guessing pretty well haha. Cheers.

[@mauriciobenjaminmossi8231]

Hi, pretty sure you are not going to see this but, at 1:22:00 you could have taken the limit of the quotient of sqrt(x) and ln(x), if the limit is infinity it means that the numerator is larger, if the limit is 0 it means the denominator is larger. Thank you so much btw. You are my main go to source for calc.

[@raffi3933]

Very smart method

[@guilherme.lameira]

In the limit comparison test the limit of the ratio of the sums cannot be 0, (59:00), otherwise you could invert Sa and Sb and get infinity instead, which wouldn't be consistent.

[@JulianVara-xm7kc]

Thank you so much! this was the video that I needed to help with the Comparison test. I was struggling so much until I found this video!

[@huangjiang6232]

the best of best and most sane Math teacher I know.

[@llaw11202]

You are an incredible instructor. I wish there were more teachers like you.

[@harshkumarlad4240]

Hands down, the Best!!!!!!!! .PERIOD

[@hawkinsallen85]

Really appreciate your lectures, Professor. Has been such a big help! Greetings from Texas!

[@freebird889]

@professor Leonard. I missed a whole week of my 5 week cal 2 course due to covid. My instructor gave me a chance to catch up, take the quizzes, and do the homework. Your videos are helping me out tremendously and I really appreciate it. Keep up the good work! Good luck on your new home!

[@ARTDEVGRU247]

5 week calc 2 course? That's insane.

[@freebird889]

@@ARTDEVGRU247 yeah. I know, but I messed up my college career so badly but just taking random classes, so by the time I figured out what I wanted to do I had to take Cal 1 and Cal 2 back to back during the summer in ten weeks. Yeah it was insane. Talk about 10-20 hours of homework a day for 10 weeks. I literally had no life lol. When we got to class, the instructor said that every day in class was like a week in a normal semester. I wanted to die the whole time.

[@ARTDEVGRU247]

@@freebird889 wow. Hopefully you did well.

[@shahlahemmati6206]

I tutor Calculus 2, and your videos on series are extremely helpful!! 🙂

[@jevcampbell2301]

I HAVE TO SAY LIKE YOU ACTUALLY GIVE ME HOPE For GRADUATING ENGINEERING. . . its so bad because like people who are actually smart but take a while to understand things... get left behind... especially in these covid times... Your videos are helping many. THANK YOU... Watching your videos last semester. SAVED ME FROM FAILING AND GOING ON ACADEMIC probation. this SEMESTER I'm passing ever test and i might not be the smartest but I'm getting better and just that feeling of getting better was all thanks to you and your videos. you not only touch. you teach how to think MATH and thats what these professors lacking.

[@zsomborczoma]

26:00 -- He writes the problem on the board --> I do the limit test and it fails (I wonder if I did it write --> copied the rest) 34:22 -- Guy asks my exact question --> Answers my question indirectly. Thank you for including the questions along with their explanations. They help a ton cause sometimes we watching, have the same questions. Also, love the fact you ask if it makes sense. Thank you!

[@clookie2004]

ten years later still out here saving lives. THIS SHOULD BE PREMIUM CONTENT PROFESSOR LEONARD DESERVES TO GET PAID

[@jeremyn9146]

My teacher is so shitty he doesn't even teach (not joking)... He just posts your videos and collects paychecks. Thanks Prof for caring for students and making math interesting and approachable.

[@JossinJax]

He clears up sequence convergence vs. series convergence quite nicely @58:00. I would just like to add that in order for the Limit Comparison to work, the limit must exist (non infinity) and must be greater than zero. REPLY

[@timetraveler1203]

your videos are literally saving my grades. Thank you so much.

[@pinestree8006]

These students are blessed with a professor like him. I do not know if this is a high school, or university, but my university's professors are nothing compared to Professor Leonard.

[@androslive]

super helpful, You're pretty much the reason I'm learnnig this material.

[@Jkarlowski2]

You are in the realest sense the only reason I'm going to make it through school. Thanks

[@yiit2925]

imagine having that teacher and telling to your friend that you have got the best calculus teacher on the planet, but have got no evidence to prove it. thank god he just visually proves that he is the best ever

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, I wish you a happy life :) I don't really know why professors van't have some patience and obedience like you

[@mehularora1813]

Idk how to even thank you, you just made my life so much easier

[@juanortegon7838]

Incredible professor. He can actually teach Calculus.

[@TheLutastic]

i gave up on my professor a long time ago I wish I knew about this man sooner. LOL

[@Algebrainiac]

I practiced this over and over and I finally understand it. Woooo!!!

[@masaart6258]

Thank uuuuuuuu I finally understand how to solve the series problems!! Your students are so lucky ☹️

[@Jaqvie]

Definitely passing my test because of Professor Leonard.

[@eudaimoniapeisithanatos2252]

I love how the class react each other, have heard a lot of bless you's there

[@rachelwalker9555]

i watch all your videos right after my lectures at school. Do you or can you post or message me the exams you give your students for organized practice material? i appreciate your thorough explanations

[@Caarve]

I agree, that would be awesome! Seems like a lot of work though. I imagine he is pretty busy

[@DynamicRoulett]

I don't think you can leak any of materials that are intended for handing out to student because of copyright and risk of plagiarisms.. Unless he has his own website that has purpose of posting lecture logs and anything related to his lecture

[@benlehrman]

Question - For limit comparison, as an example take 1/n and 1/n^2. We know 1/n diverges (harmonic) and 1/n^2 converges (p-series w/ p>1), so we'd expect to find that lim as n approaches infinity of (1/n^2)/(1/n) does not exist, but lim n -> inf (1/n^2)/(1/n) = lim n -> inf (1/n^2)*(n/1) = lim n -> inf 1/n = 0, so that limit exists, which would suggest that either both converge or both diverge but we know that not to be true. What happened?

[@KarthiLuver]

I was actually dozing off when he said "so may of you guys look like you got sleepy eyes" XD

[@restitutororbis964]

Athira Surendran Me too. Its not that the prof is boring, the topic is boring. To me convergence/divergence tests are probably the least exciting part of calculus.

[@Peter_1986]

@@restitutororbis964 My favourite calculus course is Calculus 3. Gotta love all of those colourful 3D graphs everywhere.

[@abhinavpandey6261]

its the best....I've ever learnt something new....total clarity why we doing all the steps... . . Thanks prof.

[@athershareef3225]

excellent way of teaching

[@Algebrainiac]

1:07:02 for that example I used the divergence nth term test. So if the limit is zero, series diverges. You would have to show the whole limit process and limit laws to get there but yeah that’s how I would’ve done it.

[@nintran8861]

I find this important 1:00:11 convergence for a series means that ultimately, we add up terms that are 0, coming up with a sum that is not infinity (that is an actual finite number)

[@nintran8861]

I am sometimes confused between sequence and series, especially when it comes to using limit and convergent behaviors. So this is very useful!

[@shahlahemmati6206]

Thank you so much for explaining the series in such an easy way to understand.

[@anonymousfigure37]

@41:00 instead of invoking the limit comparison test, can we make the argument that 1/(sqrt(n) + 1) >= 1/n **for all n >= 4**? and so by the comparison test the series diverges? or is stripping off terms like this for some reason not allowed? we did this with the integral test so I'm wondering why it isn't brought up here.

[@basian74]

Calc 2 at UIUC and hella glad I found this

[@Rob-gu4kd]

Anytime I don't understand something in Calc 2, I come to your videos and then I understand. You could say the limit of my knowledge diverges after watching🤯

[@Keepcalmliveon5]

I love that you state what the test is first and then ask "would you like a proof of this?" All of my professors do the proof first and I get so lost.

[@Felipe-kg8cc]

44:45 Could someone explain to me what is the significance of multiplying epsilon by L? I don't understand how it affects the proof at all. I also don't understand how multiplying epsilon by L can be true because if epsilon and L are smaller than one, it would become a smaller number and potentially smaller than |An/Bn - L|. Also for that same proof, why does Bn have to be positive? If it was negative it would flip both of the inequalities and make it so the proof still works, just the other way around.

[@HaxzProductions]

Question: Why is the comparion test difficult to use in the example at 1:08:00?

[@excelsior_academy]

Your content is an inspiration, if I can be even half as good as a teacher/tutor to my peers that'd be amazing.

[@shutnawi6635]

i'm really thankful that you've done this series (no pun intended) this is really helpful to me and others.

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your seriously the bast teacher

[@DarkCeasar85]

I think it's time I popped my KZbin comment cherry. You are the chuck norris of instructors!

[@etudemath8052]

Amazing!!! Your teaching is so nice. It is impossible to not understand. Really amazing..

[@abrown5526]

The student that says, "One more, one more, one more.." That made me laugh out loud. That is so me. Forever complaining if one more turns into another one more.

[@Peter_1986]

+Aaqilah Brown I could listen to that girl's voice all day, she sounds adorable. :D

[@rolando0151643]

Is a privilege to watch your videos, thank you a lot!!

[@Pancho674-d5f]

In the example at 1:09:40, why not just state that because series {bn} converges bc p-series (P>1) then {an} also converges by the comparison test. Why do you still move onto the limit comparison test for this example?

[@flyingpirate8115]

ı dont knoow how to thank u for these perfect teacher. these videos are my only hope for math.

[@magicmike1122]

Can you upload videos for Calc 3 and Linear algebra?

[@HunterNapier]

Took a few watches to finally start understanding the difference in what we were taking the limits and working with tests against. Ultimately realizing first we test the SEQUENCE for divergence. If that goes to zero, then we can check the SERIES for convergence. If it doesn't, you're done. Also making sure to not confuse the value of the tests with the SUM of the series (way different things). Also dont make the mistake I did and start trying to find the sum of some of these. Turns out some (like the P series) looks REALLY tough. I think in all my tests we've only been asked for Geometric (because of the formula), and maybe telescoping sums because the terms cancel out. About 58 min into this is a really good "aha" moment recap that helps explain what we're ultimately doing I think. Good luck everyone!

[@LisbackupMaj]

You are amazing. Thank you very much.

[@jordantaylor9225]

For the problem at 33:00, couldn't you do the divergence test initially and tell that it is divergent? I'm wondering why the comparison test was necessary.

[@pedrodelfino9493]

These classes are extremely high quality material