Calculus 2 Lecture 9.6: Absolute Convergence, Ratio Test and Root Test For Series

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Ratio test 25:40 Root test 1:07:00 for a quick review

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Summary: 1:20:15 As always, you excel in math and in teaching. I'm not sure if math is your passion or not, but a professor who factually has a passion for math would be hard pressed to do any better at teaching it than you, sir. Thank you very much for all the extra effort you go through to put these up on KZbin! The professor in my calc II class took less than 5 total hours of class time to go through this ENTIRE chapter right before the final exam so I am using your video series (no pun intended) to prepare for my final. This is a definite case of how a #fail or perhaps a #fallshort is corrected by a #WIN! (not a factorial)

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Intend your puns next time please.

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1:20:11 great summary

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i hate when they teach you "tricks"

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Reflexez you are realy lucky .. my teacher knows nothing about tricks in calculus.. but he is proffesional tricker to his student minds..

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@@farhatfatima1168 So he has to give you free master classes AND give you notes? We get them by writing them down.

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I'm glad you actually explain some of the more difficult algebra concepts, because most professors will neglect to do this which leaves many people confused in class.

[@abrown5526]

43:15 "n!" It's 11pm, I've been studying for hours, I'm exhausted, and that totally woke me up! Haha You're awesome Professor Leonard!

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I guess you could say *Math* Damon

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10 years later , man still a classic. You are trully that one murcielago I will never sell, thank you

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I think you should give lessons for teachers about how to teach calculus in perfect way or on good way at least :( BECAUSE YOU ARE PERFECT MR LEONARD

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Ive been taking note and have all the examples from his calculus 1 playlist up to this one so far. I'm trying to get done with the calculus 3 playlist by august.

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[@georgesadler7830]

Professor Leonard, thank you for a lengthy and well-organized video/lecture on Absolute Convergence, Ratio Test and Root Test for Series in Calculus Two. Absolute Convergence, the Ratio, Root Test and all Test for series works together in Calculus. The Chapter on Infinite Series is a tremendous amount of information for all students in Calculus Two to comprehend.

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That "Stripper joke" @ 47:20 killed me... Very nice video....it helped me a lot. Thanks prof.

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we are glad to have you. i am watching this video after 9 years and it is very useful .

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Thank you so much for uploading these videos! I learned all about series from your teachings , and I took my quiz a couple days back and got a 9.75/10. You're the best!!!

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I love your final summary in the last part of the video, it's really great!

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when you lifting at 12 but got a class to teach at one

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Is that even a sentence?

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Me too.

[@ducksoop.x]

Amazing recap at the end. I think you might have saved my semester in Calc II! I didn't do so great on the first test, and we started series and I was so lost. But you really made me understand it at an intuitive level! Thanks!

[@scalloproductions2069]

1:20:14 summary of how to use all the tests up to this section for series

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Thanks Professor Leonard! I was a bit lost about the steps and algebra around 1:03:45 but I thought it through and it makes sense now!

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watching this Dec 2021, who else raises their hands when he asks if the class understands? lol! May the Math Gods bring algorithms of praise to Prof. Leonard. Without you I would be lost. Thank You

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I've just finished my final test in calculus math .. I would like to say thank u very much.. u were so helpful..

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[@christopherwilliams9866]

54:03 Where did the -1/x^2 in the numerator come from? By the chain rule, shouldn't the derivative of ln[(n+1)/n] be 1/(n^2+n) ?

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4 years later to the date I'm watching this in preparation of my Calc 2 test

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I am super thankful for this content for free and Proffesor Leonard's dedication but in the root test there was no proof from where the root test came from right? I mean it's ok but I prefer to understand where things come from to make sure I remember them at the long run.

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[@LordAhmedAmr]

at 58:33 why didn't he treat this example as an alternating series? (I mean by that proving it's descending and the limit of (n^2*3^n)=0 )

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im passing cal 2 because of these videos

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[@gabrielllanes404]

1:43:20, I don't understand why you put sqrt(n) on the numerator for number 5, for the comparison test.

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that is exactly what I was trying to figure out ... may someone help please!!!! @professorleonard plz :)

[@foreverbored2501]

+Gabriel Llanes I don't understand that either.. But i do remember on one of his videos we proved that sq of n is larger than ln of n.. idk if u just plug in a random value that you memorize is bigger or what.. but i would love to know..

[@jubinlee2368]

+Gabriel Llanes Here's a clear solution. Of course, you can put only n as the numerator. However, what we want to use is 'p-series test', which tells us that if the infinite series has form of 1/(n^p) then it is convergent when p>1. If we take numerator as square root of n, then the right term is 1/(n^3/2) so we can apply the p-series test. As a result, we obtain that given infinite series is convergent by the comparison test!!!

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pro-tip: put the playback speed to 2x or 1.5x to save time

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it is a pro tip, do it to all my videos.

[@stephenstruble5442]

Towards the end, around 1:36:50 Why is number 5 just convergent instead of abs. convergent? Removing the negative and showing that the limit as n approaches infinity of 'a sub n' = 0 and also showing that 'a sub n' is decreasing sounds a lot like an absolute convergence test.

[@Rafiayaz]

At 19:00 Shouldn't the limits be (0

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actually the best teacher ever, my teacher is so trash

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52:48 You can use the definition of e to the "lim n-> infinity ((n+1)/n)^n" and get e instantly right??

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[@haniashehzad7136]

At 1:34:15, in the direct comparison test of b part. Isn't it should be if an diverges so bn will also diverge?

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Please cover differential equation and linear algebra.

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If you want thorough and approachable explanations of differential equations then I can highly recommend the site "Paul's Online Math Notes". He covers a lot of differential equations in his Calculus 2 section with a lot of examples, and he also covers a bunch of fundamental Linear Algebra since that's necessary for stuff like Systems Of Differential Equations etc.

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[@JohnJohnson-cr4vw]

I'm a little confused on problem ending at 56:25, if we follow the first route, then the series is divergent because e is greater than 1, however if we follow the second route, then the series is proved to be abs. convergent.... So which is correct? each route gives a different conclusion...

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He should be awarded the Noble prize for Math!!

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Respect, you are really awesome.

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48:20 REVIEW 1:30:15 EXAMPLES 1:40:00

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16:10 Absolute Convergences w/ Sin(x)

[@pedrodelfino9493]

At 56:25, professor's says the name of technique used. What is it? Is the name "the interpreter form of limit"? I am note a native speaker of English, but I am pretty watching these lessons in English. However, I can't get this right. I tried searching on Google "the interpreter form of a limit" and couldn't find anything promising. Probably I am getting it wrong.

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Pedro Delfino indeterminate form of limits. referring to L'Hopitals

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your class is amazing, will understood.

[@hectorgarciajr1693]

I got lost when Professor Leonard started to talk about factorials in the example at 55:46. Do any of you know the video, when he teaches factorials?

[@nickschneider4830]

For the example at 50:00-56:00 (approximately), when he simplified the limit down to (n^n)/((n+1)^n); could we not just say that from that step, the denominator always will be greater than the numerator given that the domain is [1, infinity), and thus, that fraction is less than 1, proving absolute convergence? Any help would be much appreciated

[@Monday97]

Nick Schneider You're correct, but some teachers would want you to complete the whole thing or they would cut some marks, but you're correct, my friend :)

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So helpful! Thank you!!

[@s.h.rafique5180]

Summary: @ 1:20:10

[@align9218]

Why is 1^(infinity) considered indeterminate? Isn't 1 to any large power always 1? 56:21

[@PartyALLNighTCrew]

Who else laughed out loud when he read "n!" as if it was an exclamation mark at 43:10 Hahaha.

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Been watching several lessons in a row... Im thinking about not going to my calculus lectures at uni from now on :D

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[@marioromero3598]

Someone divided by "0" at 17:49.

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My teacher watch videos of professor and didt teach us in the right way but in paper gives us unseen questions ... Now i find from which vdeo my teacher teaches us hope same questions she will giving to us in paper

[@alexprocel91]

At 1:43:06, I don't understand where he got sqrt(n) on the numerator. Can someone please help me on this.

[@Freakishd]

I second this

[@hakangungor4743]

41:38 the question can be solved by alternating series test too but it be convergent.However in video it abs convergent :) how can i understand which way i use help pls

[@sweetyalwal924]

At 51:19 how's that 1 it should be infinity right someone explain @Professor_Leonard