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
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.
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.
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.
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!
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
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.
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...
@@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.
He's literally so nice. His lectures are like ASMR, as in his delivery, his humor, his niceness - genuine concern whether his students understand, his super detailed explanations. I laugh when he makes a joke and instinctively raise my hand whenever I get a problem. I love how he goes over basic algebra becaue I'm not too fast, like I can't see the steps in my head.
He is probably the best teacher at math I have ever seen. 100X better than my calc 2 professor now. This is definitely my 'go to' for calc 2. Maybe I will get higher than a 40% for my next exam learning from him instead of my professor now...
I hope to become a math lecturer one day. Professor Leonard is such an inspiration.. He explains very well and seems to have a real passion for teaching in general. What a legend!
My professor goes extremely fast, you explain these concepts so well and very slowly to ensure understanding. Thank you for uploading these videos!!! I was beyond lost, 1 hour and 30 minutes into this lecture and everything is so clear to me. Thank you Professor Leonard!!
Professor Leonard you are fantastic. I'm seriously starting to doubt the quality of education I'm currently paying for at Iowa State University after seeing what a great professor is like and what I have been missing.
Yeah, same here. My current professor is terrible, his voice is extremely monotone and it feels like he is talking to himself, and he almost never interacts with the students. He doesn't even look bored or annoyed, he just has a completely blank face. When I attend his lectures I am just like "wtf are you saying?" the whole time, lol.
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!
Professor Leonard is an awesome supplement to my class. My teacher is pretty good, but I often need to listen to it a second time, so i just put it on 2 times the speed and I understand that material fairly well.
You are really gifted in the field of lecturing PROFESSOR LEONARD. I rank you above all the mathematics teachers I've seen so far. I feel extremely happy anytime I learn from your lessons. Only God can reward you for your contributions. I wish you the best in the world and best of guidance for the success of the hereafter. wish you all the best and I can't stop thanking you so much. welldone SİR.
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.
Your videos are a blessing. I don't care for my calc teacher too much, but I really enjoy your videos. You're engaging and you explain the why and how much better. I really appreciate it.
You are a fantastic professor. Thank you so much for posting these, they are extremely helpful. I wish more professors had your passion and skill for teaching.
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!
This was excellent. I've been a section or 2 behind in my calc class for half the semester because my prof tends to rush through the material and skip steps. Love the way you break things down little by little and repeat important points
Your videos have allowed me to learn a lot of calculus 2 material before I've even started the class. Just wanted to say thank you! You're a great teacher.
Thank you SO much for sharing this video with the world's Calculus students. It really helps to have someone clearly explaining these concepts at a reasonable pace.
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.
Difference between you and my current professors: They only teach me how to do something, you help me understand WHY I'm doing something and THEN show me how to do something properly. Your lessons always stick with me, Professor Leonard, other professors' lessons never do! They always go right over my head. Thank you for what you do!
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.
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!
Awesome, He is a Rock Star in teaching Mathematics.!! I become a fan for him now.. he just teaches complex stuff and makes us understand on the first go.., unbelievable Man. Lots of respect , because u just teaches the subject in the most beautiful way and NOT show off the knowledge gained ( most average professors do this mistake.)
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!
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.
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
+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.
I have learned so much from Professor Leonard. I believe he is the best teacher in the world. My college teacher was so worst and I could not pass the exam, so I could not sleep and was going to die.In the hospital, I got Leonard's lecture.His lecture saved my lecture. He is not only my good teacher but also my life.Now I am having a happy life and math is so easy for me.Anyway my major is math.
I am a senior in high school preparing for the AP Calculus BC exam, which is the equivalent of Calculus II by myself and I couldn't get any better help than your videos, thank you so much!!
I'm retaking calc 3 on this upcoming spring and all these god blessing videos are really helping me out to prepare for passing course this time. As always, thank you for filming and posting such a wonderful lectures. Seriously god bless you
THANK YOU SO MUCH FOR ALL THE VIDEOS IN THE COURSE! They have been so helpful for me to understand all the material of Calculus 2! You are really good at explaining and it is easy to see how dedicated you are to your job and your students!!
Professor Leonard, you are a college student life saver lol. I mean your lectures are sooooooo clear. Thank you for taking your time to post these videos.
4:45 is the reason I've watched all your videos, you SUM up (pun intended) information and recap to make sure we follow! thanks so much! cheers from Montreal, Canada
Just stumbled on this playlist half way through a self directed online course through Matlab. Wish I knew about it sooner. Great teacher. Already looking forward to watching these lectures for calc 3 and diff eq. Props
Thank you very much , i always feel good after watching your videos and it seems tht i will get A+ on Calculus , i wish i have such a professor at my university))
Thank you.. I've spent countless hours reading the calc textbook and learned nothing. And when I watched your video, I understand it right a way! Thank u! :) Your awesome! I wish your our teacher!
Introduction to Series 0:00 Steps for solving Infinite Series: 19:20 Telescoping Series 27:20 Geometric Series 51:20 Harmonic Series 1:28:03 Divergence Test 1:40:00 Properties of Convergent Series 1:42:11 Divergence Test again: 1:51:08
I cannot express enough the magnitude of my gratitude to you. Thanks. The only thing I hate are the jelly-bellies answering the simplest of your questions AFTER you had given the correct answer and had repeated it twice.
My caluculus proffessor is teaching this stuff so fast that i can't understand anything. Thanks to you Prof. Leonard, i understand what calculus is about. Please do more videos.
Dude, I die at that student's commentary @ 13:30 😂 "Take a leg" idk if that's a saying, but I totally agree. He's so engaged w/ the math like, "Listen up... you guys 'bout to see something cool " lool But I give him props cos he explains it mad simple to actually be amazed by math 😅 Great job Prof Leonard!
People like yourself should be offered a doctorate in teaching or something similar. You offer way much more than most that have a PhD. I understand that a PhD has a lot to do with research, but you have clearly researched and put a very popular and successful teaching method into practice.
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....
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!!!
My Calc two teacher is very theoretical, he's great don't get me wrong, but I have to first watch Professor Leonard's videos to understand my current teacher's videos. I've been watching Professor Leonard's videos since before calculus. Excellent teacher.
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
"Gauss was super smart. Now he's DUMB, 'cause he's dead, right?" Haha I almost spewed my coffee all over my computer and books.I love how Professor Leonards's humor makes learning this so much more entertaining.
1:13:05 Now that I'm on the right track, I find it hilarious how my Calculus lecturers insist that a valid sum exists even if a series is divergent... I knew something was off! Thanks Leonard! Privileged to be in your "virtual" class!
Math notation can be displayed at: asciimath.org/ 00:00:00 INFO: Series Explanation *) NOTATION: sum_(n=x_1)^(x_2) a_n, where x_1 is the starting point & x_2 is the final term *) x_1 >= 0 *) A series is when you add up ALL of the terms from a sequence sum_(n=1)^(∞) a_n *) A partial sum is when you add a finite number of terms from a sequence sum_(k=1)^(n) a_k = S_n *) For every subsequent n will yield a new S_n *) If {S_n} converges, lim_(n->∞) S_n = S *) If {S_n} diverges, sum_()a_n diverges 00:19:50 INFO: Series Example 1 *) Step 1: Find partial sum *) Step 2: Find formula for S_n *) Step 3: Take lim_(n->∞) S_n *) Step 4: Connect the dots: The sequence of partial sums is divergent ∴ series is divergent 00:27:20 INFO: Telescoping Series Example 1 *) Find partial sum by plugging in n *) Write out your first few terms *) Write out your final terms by writing them as a notation of n *) Determine how the subsequent terms cancel each other out to find S_n *) Take lim_(n->∞) S_n and determine if it converges 00:41:20 INFO: Telescoping Series Example 2 *) Rewrite series as two fractions *) Solve partial fractions *) Solve partial sum; write first few then final terms in notation of n *) Determine was cancels *) Connect the dots: Take lim_(n->∞) S_n and determine if it converges 00:51:20 INFO: Geometric Series Explanation *) sum_(n=1)^(∞)ar^(n-1) where a = constant, r = base, raised to some power n + 1 *) sum_(n=1)^(∞)ar^(n-1) = a + ar^1 + ar^2 + ar^3... *) Series will converge when |r| < 1 or -1 < r < 1 *) Series will diverge when |r| >= 1 *) Partial sum of series can be derived as sum_(n=1)^(∞)ar^(n-1) = a / (1-r) 01:09:39 INFO: Geometric Series Example 1 *) Determine if series is a geometric series? -> Is the first exponent of the series 0? *) Identity your a & r *) Plug it in to: a / (1-r) = S_n 01:12:40 INFO: Geometric Series Case Examples *) Determine that the series is convergent first, then find the sum 01:14:13 INFO: Geometric Series Example 2 *) Formatting series into a familiar geometric series format 01:21:50 INFO: Geometric Series Example 3 *) Manipulating series into familiar geometric series format 01:27:45 INFO: Harmonic Series Explanation *) Harmonic series: sum_(n=1)^(∞) 1/n = 1 + 1/2 + 1/3 + 1/4 .... ∞ *) Harmonic series is divergent *) If sequence {a_n} is convergent, any sub sequence must also be convergent *) To show {a_n} diverges, it is enough to show a sub sequence diverges *) If a series sum_(n=1)^(∞) a_n converges, lim_(n->∞) a_n = 0 *) IF lim_(n->∞)a_n = 0, DOES NOT NECESSARILY MEANS CONVERGENCE *) If lim_(n->∞) a_n ≠ 0 or D.N.E the series sum_(n=1)^(∞) a_n diverges 01:41:10 INFO: Divergence Test 01:42:10 INFO: Properties of Convergent Series *) Constant rule: sum_(n=1)^(∞) C*a_n = C*sum_(n=1)^(∞) *) Sum/difference rule: sum_(n=1)^(∞) (a_n +- b_n) = sum_(n=1)^(∞) a_n + sum_(n=1)^(∞) b_n *) Fraction rules sum_(n=1)^(∞) (2^n-5^n)/3^n = sum_(n=1)^(∞) [(2^n/3^n) - (5^n/3^n)], both fractions need to be convergent
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
I love you :D
Life saver!
I love your videos on Cities Skylines! I was surprised to see you on a math video like this lol.
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.
Much appreciated. Especially since I'm having a really tough time with this subject.
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.
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...
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.
Your teacher gets mad when you ask questions? He/she should leave immediately and never return to teaching
He's young... let him continue teaching for a couple more decades, he'll tire out and be like you professor
@@akashPatel-sf2eo Then don't teach at that point, bruh.
Shoutout to the classroom for also being confused when I'm confused.
1:07:55 I think someone said "1" for "a*1"
@@andrewlachance2062😂
😂😂😂😂😂huh crazy
@@andrewlachance2062 You didn't have to call them out like that lol
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!
You're Welcome!! Thanks for watching and enjoy :)
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
you forgot the telescoping series at 41:45
There's a telescoping series example that starts around 30 minutes mark (go to 30 mins its that example)
you are like Jesus but better
I disagree bro, Jesus is way much better ;)
@TMTA ` why do you think like this?
Not only are Leonard's videos a godsend but so is the replay button.
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.
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...
@@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.
@@adabujiki
99.999% of series do NOT have a simple formula, it is not winging.
Please live forever. You helped me so much and I can only imagine the good you would do with an extra hundred years or so.
Linc DeRose his videos will live forever
Minato Namikaze we need him to live forever for gracing us with his knowledge!
He's literally so nice. His lectures are like ASMR, as in his delivery, his humor, his niceness - genuine concern whether his students understand, his super detailed explanations. I laugh when he makes a joke and instinctively raise my hand whenever I get a problem. I love how he goes over basic algebra becaue I'm not too fast, like I can't see the steps in my head.
He is probably the best teacher at math I have ever seen. 100X better than my calc 2 professor now. This is definitely my 'go to' for calc 2. Maybe I will get higher than a 40% for my next exam learning from him instead of my professor now...
I hope to become a math lecturer one day. Professor Leonard is such an inspiration.. He explains very well and seems to have a real passion for teaching in general. What a legend!
got a 100% on my first calc II exam,, first time taking a math course in nearly 10 years,, thanks professor leonard! owe it to you
I actually raise my hand every time you say so..
Hajar x3 😂😂😂 so I'm not the only one
Hajar x3 same here
same here lol
I said bless you when someone in his class sneezed and my friends looked at me like I'm crazy lol
I answer when he asks if they understand.
I honestly don't get how you make this so easy to understand.
He is a good Professor :)
My professor goes extremely fast, you explain these concepts so well and very slowly to ensure understanding. Thank you for uploading these videos!!! I was beyond lost, 1 hour and 30 minutes into this lecture and everything is so clear to me. Thank you Professor Leonard!!
Professor Leonard you are fantastic. I'm seriously starting to doubt the quality of education I'm currently paying for at Iowa State University after seeing what a great professor is like and what I have been missing.
I don't go to lectures, I only watch Professor Leonard
Yeah, same here.
My current professor is terrible, his voice is extremely monotone and it feels like he is talking to himself, and he almost never interacts with the students.
He doesn't even look bored or annoyed, he just has a completely blank face.
When I attend his lectures I am just like "wtf are you saying?" the whole time, lol.
It has become a waste of gas for me. I just watch the lectures here and pay my school to test me and give me a piece of paper.
@@Peter_1986 sameeeeeeee ı totally agree with u
Im in calc lecture now, and here I am,,
@@Peter_1986 SAAAAAAAMMMMMEEEEEE :)
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!
You're truly a blessing. I hope your videos never get taken down
Professor Leonard is an awesome supplement to my class. My teacher is pretty good, but I often need to listen to it a second time, so i just put it on 2 times the speed and I understand that material fairly well.
You are really gifted in the field of lecturing PROFESSOR LEONARD. I rank you above all the mathematics teachers I've seen so far. I feel extremely happy anytime I learn from your lessons. Only God can reward you for your contributions. I wish you the best in the world and best of guidance for the success of the hereafter. wish you all the best and I can't stop thanking you so much. welldone SİR.
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.
If I want step-by-step, thorough explanations, this is the channel I go to! Thanks!
I wish I would have found your channel sooner! you make everything soooooo clear!!!! THANK YOU!!!!
I did not get him when I took calc 1 but he saved my life for calc 2.
Your videos are a blessing. I don't care for my calc teacher too much, but I really enjoy your videos. You're engaging and you explain the why and how much better. I really appreciate it.
You are a fantastic professor. Thank you so much for posting these, they are extremely helpful. I wish more professors had your passion and skill for teaching.
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!
I'm very glad they are helping!!
PROFESSOR LEONARD IS THE BEST, HOPE ALL THE LECTURES WOULD JUST LEARN FROM HIM.
This was excellent. I've been a section or 2 behind in my calc class for half the semester because my prof tends to rush through the material and skip steps. Love the way you break things down little by little and repeat important points
Your videos have allowed me to learn a lot of calculus 2 material before I've even started the class. Just wanted to say thank you! You're a great teacher.
Thank you SO much for sharing this video with the world's Calculus students. It really helps to have someone clearly explaining these concepts at a reasonable pace.
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.
Difference between you and my current professors: They only teach me how to do something, you help me understand WHY I'm doing something and THEN show me how to do something properly. Your lessons always stick with me, Professor Leonard, other professors' lessons never do! They always go right over my head. Thank you for what you do!
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.
YOU ARE THE KING OF CALCULUS.. THANK YOU SO MUCH FOR THIS EXPLANATION PROFESSOR LEONARD!!!!!!!!
Thank God you exist. Thank you so much for the knowledge, Professor.
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!
professor...please..please...teach calc 3. and discrete. and linear algebra.
+Z Dela Cruz YES i need to find a good youtube professor to teach Discrete and Linear algebra!!!!
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.
+Al Casablancas WHAT SCHOOL
+Colorado DayDreams those are both 2 of my favorite classes lol
Merced Community College
While I never had him as a professor in real life, still one of the best profs I've experienced thus far
You are a gentleman and a scholar.
So are you kind sir
شكرا مستر ليونرد
thanks Professor Leonard
.
توووووبي
Awesome,
He is a Rock Star in teaching Mathematics.!! I become a fan for him now..
he just teaches complex stuff and makes us understand on the first go.., unbelievable Man.
Lots of respect , because u just teaches the subject in the most beautiful way and
NOT show off the knowledge gained ( most average professors do this mistake.)
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!
I'm doing an independent study of Calc II and these videos are immensely helpful!
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.
I genuinely appreciate that complement. Thank you very much. Enjoy Real Analysis!!!
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
+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.
I have learned so much from Professor Leonard. I believe he is the best teacher in the world. My college teacher was so worst and I could not pass the exam, so I could not sleep and was going to die.In the hospital, I got Leonard's lecture.His lecture saved my lecture. He is not only my good teacher but also my life.Now I am having a happy life and math is so easy for me.Anyway my major is math.
mate you scared the shit outta us with that smile after "I'M NOT GONNA LIVE FOREVER, OR AM I?"
I am a senior in high school preparing for the AP Calculus BC exam, which is the equivalent of Calculus II by myself and I couldn't get any better help than your videos, thank you so much!!
You're the man Professor Leonard! Your classes are so much clearer than mine at my University. Thanks for sharing!
You are the best teacher I have ever seen...thanks for your free service... please keep teaching like this for years and years...thanks
I'm retaking calc 3 on this upcoming spring and all these god blessing videos are really helping me out to prepare for passing course this time. As always, thank you for filming and posting such a wonderful lectures. Seriously god bless you
You really put your heart into these lectures.
Just scored 75% in my last Calculus II exam, a very hard one, half the class zero'd it, in large part thanks to you. God bless.
I love that the partial fraction decomposition turns perfectly into a telescoping series. Brilliant!
I can't thank you enough professor Leonard for making me love calc 2. i love the full detailed steps. thank you again.
THANK YOU SO MUCH FOR ALL THE VIDEOS IN THE COURSE! They have been so helpful for me to understand all the material of Calculus 2! You are really good at explaining and it is easy to see how dedicated you are to your job and your students!!
I have a Calc 2 exam on Monday and have really fallen behind, thanks so much for these videos they really explain the topics so much better!
Professor Leonard, you are a college student life saver lol. I mean your lectures are sooooooo clear. Thank you for taking your time to post these videos.
You are saving my university grades. THANK YOU!!!
You are an amazing teacher. I have college professors who do not explain these math concepts as well as you do. Thank you for posting your lectures.
your TH-cam videos will live forever!!!!! and thus you as well!
Thank you professor Leonard, I've been watching your videos all through calc 1 and now 2.
Thank you Professor Leonard. I do not know how I will do on my test, but I feel really good about Calculus 2 because of you. Thank you so much.
4:45 is the reason I've watched all your videos, you SUM up (pun intended) information and recap to make sure we follow! thanks so much! cheers from Montreal, Canada
Just stumbled on this playlist half way through a self directed online course through Matlab. Wish I knew about it sooner. Great teacher. Already looking forward to watching these lectures for calc 3 and diff eq.
Props
Just finished my calculus 2 exam , it was easy like a piece of cake thank you professor really .
Best wishes from Palestine
Best wishes to you and all Palestine from California
Thank you very much , i always feel good after watching your videos and it seems tht i will get A+ on Calculus , i wish i have such a professor at my university))
did this video help you get an A+ ?
Went from understanding nothing to feeling like I mastered this topic thanks so much
Thank you.. I've spent countless hours reading the calc textbook and learned nothing. And when I watched your video, I understand it right a way! Thank u! :) Your awesome! I wish your our teacher!
Introduction to Series 0:00
Steps for solving Infinite Series: 19:20
Telescoping Series 27:20
Geometric Series 51:20
Harmonic Series 1:28:03
Divergence Test 1:40:00
Properties of Convergent Series 1:42:11
Divergence Test again: 1:51:08
I cannot thank you enough for these videos I swear!
I cannot express enough the magnitude of my gratitude to you. Thanks. The only thing I hate are the jelly-bellies answering the simplest of your questions AFTER you had given the correct answer and had repeated it twice.
My caluculus proffessor is teaching this stuff so fast that i can't understand anything. Thanks to you Prof. Leonard, i understand what calculus is about. Please do more videos.
Thank you for the thoughtful and articulate instructions. I feel like I will be able to pass Calculus 2 with the help of your videos!
Prof. Leonard Everything make sense to us, thank you very much
Dude, I die at that student's commentary @ 13:30 😂 "Take a leg" idk if that's a saying, but I totally agree. He's so engaged w/ the math like, "Listen up... you guys 'bout to see something cool " lool But I give him props cos he explains it mad simple to actually be amazed by math 😅 Great job Prof Leonard!
I had so much trouble with telescoping series until watching this video. Thanks for explaining everything so clearly!
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.
Professor Leonard is really really really the best teacher in the world. And he has saved my life and dream
People like yourself should be offered a doctorate in teaching or something similar. You offer way much more than most that have a PhD. I understand that a PhD has a lot to do with research, but you have clearly researched and put a very popular and successful teaching method into practice.
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....
Thank you for these amazing videos. I wish more professors taught like you.
Finals is NEXT WEEK, THIS IS A BIG HELP
Have an exam in about 4 hours. I think you just saved my grade! Thank you!
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!!!
My lectures are cancelled due to coronavirus, so this is SUPER needed thank you Professor Leonard
My Calc two teacher is very theoretical, he's great don't get me wrong, but I have to first watch Professor Leonard's videos to understand my current teacher's videos. I've been watching Professor Leonard's videos since before calculus. Excellent teacher.
Thanks a lot Leonard, you explain everything in a very understandable way.
Great teaching! Your videos are long, but well worth the time. Thank you.
professor you are an amazing person and I appreciate all of your hard work
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
Love these lectures; they give you a good intuitive grasp of mathematics! Just what you need when you are a physics student!
Mr Leonard deserves more subscribers. Good work!!!
Love your classes! You're an awesome teacher!
"Gauss was super smart. Now he's DUMB, 'cause he's dead, right?" Haha I almost spewed my coffee all over my computer and books.I love how Professor Leonards's humor makes learning this so much more entertaining.
1:13:05 Now that I'm on the right track, I find it hilarious how my Calculus lecturers insist that a valid sum exists even if a series is divergent... I knew something was off!
Thanks Leonard! Privileged to be in your "virtual" class!
look up ramanujan summation and zeta function regularization
My series bewilderment just vanished. A massive thank you, prof Leonard.
thank you Professor Leonard
Math notation can be displayed at: asciimath.org/
00:00:00 INFO: Series Explanation
*) NOTATION: sum_(n=x_1)^(x_2) a_n, where x_1 is the starting point & x_2 is the final term
*) x_1 >= 0
*) A series is when you add up ALL of the terms from a sequence sum_(n=1)^(∞) a_n
*) A partial sum is when you add a finite number of terms from a sequence sum_(k=1)^(n) a_k = S_n
*) For every subsequent n will yield a new S_n
*) If {S_n} converges, lim_(n->∞) S_n = S
*) If {S_n} diverges, sum_()a_n diverges
00:19:50 INFO: Series Example 1
*) Step 1: Find partial sum
*) Step 2: Find formula for S_n
*) Step 3: Take lim_(n->∞) S_n
*) Step 4: Connect the dots: The sequence of partial sums is divergent ∴ series is divergent
00:27:20 INFO: Telescoping Series Example 1
*) Find partial sum by plugging in n
*) Write out your first few terms
*) Write out your final terms by writing them as a notation of n
*) Determine how the subsequent terms cancel each other out to find S_n
*) Take lim_(n->∞) S_n and determine if it converges
00:41:20 INFO: Telescoping Series Example 2
*) Rewrite series as two fractions
*) Solve partial fractions
*) Solve partial sum; write first few then final terms in notation of n
*) Determine was cancels
*) Connect the dots: Take lim_(n->∞) S_n and determine if it converges
00:51:20 INFO: Geometric Series Explanation
*) sum_(n=1)^(∞)ar^(n-1) where a = constant, r = base, raised to some power n + 1
*) sum_(n=1)^(∞)ar^(n-1) = a + ar^1 + ar^2 + ar^3...
*) Series will converge when |r| < 1 or -1 < r < 1
*) Series will diverge when |r| >= 1
*) Partial sum of series can be derived as sum_(n=1)^(∞)ar^(n-1) = a / (1-r)
01:09:39 INFO: Geometric Series Example 1
*) Determine if series is a geometric series? -> Is the first exponent of the series 0?
*) Identity your a & r
*) Plug it in to: a / (1-r) = S_n
01:12:40 INFO: Geometric Series Case Examples
*) Determine that the series is convergent first, then find the sum
01:14:13 INFO: Geometric Series Example 2
*) Formatting series into a familiar geometric series format
01:21:50 INFO: Geometric Series Example 3
*) Manipulating series into familiar geometric series format
01:27:45 INFO: Harmonic Series Explanation
*) Harmonic series: sum_(n=1)^(∞) 1/n = 1 + 1/2 + 1/3 + 1/4 .... ∞
*) Harmonic series is divergent
*) If sequence {a_n} is convergent, any sub sequence must also be convergent
*) To show {a_n} diverges, it is enough to show a sub sequence diverges
*) If a series sum_(n=1)^(∞) a_n converges, lim_(n->∞) a_n = 0
*) IF lim_(n->∞)a_n = 0, DOES NOT NECESSARILY MEANS CONVERGENCE
*) If lim_(n->∞) a_n ≠ 0 or D.N.E the series sum_(n=1)^(∞) a_n diverges
01:41:10 INFO: Divergence Test
01:42:10 INFO: Properties of Convergent Series
*) Constant rule: sum_(n=1)^(∞) C*a_n = C*sum_(n=1)^(∞)
*) Sum/difference rule: sum_(n=1)^(∞) (a_n +- b_n) = sum_(n=1)^(∞) a_n + sum_(n=1)^(∞) b_n
*) Fraction rules sum_(n=1)^(∞) (2^n-5^n)/3^n = sum_(n=1)^(∞) [(2^n/3^n) - (5^n/3^n)], both fractions need to be convergent
Damn why is Professor Leonard so good at explaining stuff. High quality video as always!
THANK GOD YOU EXIST !