Calculus 2 Lecture 9.2: Series, Geometric Series, Harmonic Series, and Divergence Test

[@Fluxtrance]

If you're like me and reviewing for an exam, here are some time stamps! Steps for solving Infinite Series: 19:20 Geometric Series: 51:20 Harmonic Series: 1:28:03 Divergence Test: 1:51:08

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Life saver!

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[@riccardoossanna8170]

You need to find the partial sum (or a pattern for it) to calculate its limit for "k" tending to infinity. Therefore, if this limit exist and is finite ( let's call it S), it will be the same number the sum of "n" (or series) will be going to. Obviously, if the partial sum Sn goes to infinity the series will diverge.

[@ducksoop.x]

Much appreciated. Especially since I'm having a really tough time with this subject.

[@Corypro21]

Between his clever humor, delivery/ enunciation, and clear concise breaking down of steps. This man deserves an award. At least from youtube or something. My calculus 2 teacher is the COMPLETE OPPOSITE of Prof. Leonard. Gets mad when you ask questions and doesn't explain things clearly and doesn't seem to care about the well-being of us students. As an aspiring math teacher to be, you are a great role-model and I hope I can be half the teacher you are.

[@desC0D3]

My professor is the same way. Seriously, why don't they explain as Prof Leonard? My only assumption is that they my deem it less sophisticated...

[@tahirrao5173]

Someone that gets angry when asked about the subject that they are supposed to teach is perhaps not passionate about the work that he or she is doing. It is may be just a job and not the work that they probably respect and idealize themselves.

[@VergilPeterson]

Your teacher gets mad when you ask questions? He/she should leave immediately and never return to teaching

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He's young... let him continue teaching for a couple more decades, he'll tire out and be like you professor

[@paladin1147]

@@akashPatel-sf2eo Then don't teach at that point, bruh.

[@elihodges5593]

Shoutout to the classroom for also being confused when I'm confused.

[@andrewlachance2062]

1:07:55 I think someone said "1" for "a*1"

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[@heshv.m]

@@andrewlachance2062 You didn't have to call them out like that lol

[@hazimrassam9389]

Introduction to Series 0:00 Telescoping Series 27:20 Geometric Series 51:20 Harmonic Series 1:28:05 Divergence Test 1:40:00 Properties of Convergent Series 1:42:11 The rest is just more examples

[@Cristobal512]

you forgot the telescoping series at 41:45

[@hasaniqbal233]

There's a telescoping series example that starts around 30 minutes mark (go to 30 mins its that example)

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[@6ilchrist]

My college professor is a very nice person but presents material in a very disorganized manner. I've been struggling through Calc 2 by watching youtube videos, reading the book, and visiting Khan Academy and your lectures have been such a huge help. I just wanted to say thank you. Thank you!

[@ProfessorLeonard]

You're Welcome!! Thanks for watching and enjoy :)

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[@thebassofmontecristo]

There is nothing so valuable to math understanding like a Socratic, conversational style of teaching. Personally, I don't think I would have ever understood series if it wasn't taught the way it was on this video. The first 17 minutes of this video were eye-opening to me and once I saw WHY the existing (converging) limit of a partial sum HAS to be the actual sum of an infinite series I stood in awe of the paradoxical simplicity and depth of this topic. This is the beauty of numbers at work. Professor Leonard, you just added to my love of mathematics today. Thank you.

[@adabujiki]

yeah but at the 30th minute his argument starts to fall apart... he no longer can develop a formula for "Sn" using a proven method... he starts to wing it lol... still great guy...

[@hasaniqbal233]

@@adabujiki he does use a proven method though, its the telescoping series method. He's writing out the sum of each term of the series (ie the partial sum) knowing that because it's a sum/difference of fractions, most of the terms will cancel out and leave an easy sequence which's limit can be taken.

[@Antonio_Serdar]

@@adabujiki 99.999% of series do NOT have a simple formula, it is not winging.

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I answer when he asks if they understand.

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For those that don't get it the 1st time, you may wanna watch it again. Definitely made more sense the 2nd time for me. Good luck to all and THANK YOU PROFESSOR LEONARD!

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[@kodya4]

Series and sequences have been the first time studying math where I have broke down from pure frustration. Thank you for taking the time to teach us this the way it should be taught!

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[@georgesadler7830]

Professor Leonard, thank you for an amazing video/lecture on the Introduction to Series, Geometric, Harmonic Series and the Divergence Test in Calculus Two. Pattern recognition is important when analyzing series in Mathematics. I also encountered series in Digitel Signal Processing and Signal and Systems Theory.

[@2cool607]

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Thank you so much for these videos! I love mathematics and I've always loved them since little but since the introduction of limits, maths started taking a downturn for me due to the surge of mental issues and my current teacher, who acts entitled in every way shape and form, doesn't help the fact I'm struggling. Still, I find your videos to be a salvation, really! You present this in such a friendly and caring manner, interacting with the classroom and so on, it feels like I'm back in my first 'highschool', which is called Secondary School in my country, year once again and I just love listening to these in the background as I do exercises or take a break from solving them to clear my thoughts and review the topics I just practised. Thank you so much for these! They are helping me through really hard times!

[@ProfessorLeonard]

I'm very glad they are helping!!

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I love that the partial fraction decomposition turns perfectly into a telescoping series. Brilliant!

[@jayvondied5385]

Went from understanding nothing to feeling like I mastered this topic thanks so much

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Thank you for making these videos. You've made series make sense for me and helped me pass my Calc 20B class. I can't express my thanks enough for you putting these videos up for others to see.

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[@kimichh183]

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Awesome work on this video! I wish I had more professors like you. You make calculus fun and engaging, which is... difficult, to say the least. The Calc II professor in my 3 1/2 hour night class is a complete snore, and I don't learn half of what I learn from you. Honestly, I don't understand why your students don't laugh at half of your jokes. I thought they were hilarious! Whether they react or not though, you should know that you infuse a fairly dry course with just the right amount of juiciness. Don't change a thing about your style.

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Missed a lesson in Calc 2 the other day on Series, and I'm so glad I remembered this videos! I'm already s refreshed, and learning much more than I do in class! Thanks a bunch!

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I do not know how to say that but I wanna thank you I just saw my calc 2 grade and I passed the class.You are on of the reason that made me pass the class. thak you very very much professor you are the best!!!

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This is a really high quality presentation of series. Thanks a lot -- and I'm using this video for a real analysis class, which is saying something about the quality of your treatment of this topic.

[@ProfessorLeonard]

I genuinely appreciate that complement. Thank you very much. Enjoy Real Analysis!!!

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Yeah truly! I would rather watch these videos then attend my Calculus II lectures. Seems like a waste of time compared to the presentation of all these lessons

[@Peter_1986]

+Professor Leonard I am using these videos for my "Linear Analysis" class. I am not sure if there is quite a course like that in America, but in Sweden that course is chiefly about the Fourier Series and the Fourier/Laplace transforms (which, of course, means that we start with Sequences and Series), and also some stuff about systems of linear differential equations, which I find pretty fascinating since it's kind of like a blend of Linear Algebra and Differential Equations.

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I had so much trouble with telescoping series until watching this video. Thanks for explaining everything so clearly!

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you´ll have a harder time finding sources for ODEs and PDEs and linear algebra, so far ive found no source for those, only khan academy.

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professor...please..please...teach calc 3. and discrete. and linear algebra.

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oh cool. i heard he teaches all these courses, he just doesn't post them online. hopefully, he post his calc 3 spring sem. online.

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Finals is NEXT WEEK, THIS IS A BIG HELP

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Thank you so much, you are amazing. My teacher had notes saying that a series is a bunch of partial sums but we didn't go into why. I come to your videos and you have all the answers. You are amazing

[@dheibeljr]

Professor Leonard I love you so much, with the virus my professor (who is a great guy) is teaching us this via video conference typing up equations and series USING ONENOTE! IT JUST DOESN'T WORK!!!! I get practically to the point of tears during each lecture because all he can do it type up something and talk, he cannot type up all the steps and supporting info needed like you do or it would take forever and of course there is always the one kid who understands everything always pushing the professor to go faster while it is clear everyone else is suffering and lagging behind but because it is such a poor platform for teaching math of all things he has no choice but to move on or we would never finish the material on time. All our tutoring is the same way and ineffective while also having to be scheduled way out in advance (as if we know when we will not understand something ahead of time) due to everything going online, I know I can do this material if taught properly and given the chance to actually learn it and process things but man it is rough, really rough, with the absurd way we are being taught online is the main downfall! Your videos are literally a god send! THANK YOU!!!!

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Professor Leonard, you're an excellent professor and I'm so thankful for your videos! It's sad that I learn more from you than from the university I pay thousands to attend....

[@wealthynanaj]

My series bewilderment just vanished. A massive thank you, prof Leonard.

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Don't get me wrong I think my professor is great and is good at lecturing my classmates and me, but there is just something in my brain that has trouble putting things together since he is not the best at english. Having someone with crisp and clear pronunciation helps out so much and I am grateful you posted these videos online!

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phenomenal. Thank you Professor Leonard for "dumbing it down" far enough for even a d.a. like myself to understand! Makes it look easy!

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did this video help you get an A+ ?

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I'd consider myself to be someone with a slight attention span problem, but I was able to watch this is a solid 2 sittings & I understand so much now!!

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