+ViaVanille I have a physics teacher in Wave Mechanics who is a lot like Leonard - he has an interesting habit of spending the last few minutes right before a lecture begins on playing random live shows of heavy metal bands for some reason, probably just to entertain us while we wait for class to begin, and he also makes a lot of references to Star Wars in a lot of his problems (he has even dressed up as Han Solo in some exam problems and made himself a character in those problems, lol). He also has a rather awesome attitude to his computer during the classes - whenever it refuses to cooperate he is like "what a piece of crap =P", or casually asks the students for help with it. He is always in a great mood as well, and clearly enjoys his job. It is teachers like him and Leonard who make lectures fun.
00:00:00 INFO: Integral Test Explanation *) f(x) = f(n) = a_n *) f(n) must be continuous, positive & decreasing *) If integral converges, so will series *) Does not always provide sum of series 00:10:36 INFO: Integral Test Example 1 *) Step 1: Identify f(x) *) Step 2: Determine if it is positive, continuous & decreasing *) Step 3: Solve improper integral *) Step 4: Connect the dots: lim f(x) exist ∵ series is convergent 00:19:40 INFO: Integral Test Example 2 00:26:27 INFO: Integral Test Example 3 *) Proving f(x) = ln(x)/x is decreasing *) The integral diverges ∵ series is divergent 00:41:08 INFO: P Series Explanation *) P series will always diverge when P 1 00:47:20 INFO: P Series Summary 00:49:38 INFO: P Series Example 1 00:51:00 INFO: P Series Example 2 *) Root P 00:52:00 INFO: P Series Example 3 *) Negative P 00:57:45 INFO: Integral Test Example 4 *) Will divergence test work? No *) Is it a geometric series? No *) Is it a p series? No *) Can you use Integral Test? Yes, f(x) is positive, continuous & decreasing
God knows how many good teacher qualities I saw in him like explaining so amazingly ,not assuming we'll know-it-all instead explaining every step , asking short questions for maintaining attention of students ,complimenting (like good ,...) for correct answers, being patient and calm,making jokes and light environment in class...I am blessed to watch this!!!!!!! THANK YOU SIR SOOOO SOOOOO MUCH.
Raquel Gallo Hello, I didn't elaborate because this particular example was covered in section 7.6 at 43:15. Please check it out! Basically, the divergence test for the example you are referencing here shows divergence for any "p" = 0 or "p" < 0. For "p" = 1, we create a Harmonic Series (Divergent). For "p" between 0 and 1, we use the example of an improper integral to show that the function of 1/x^p (I believe I use x^n in 7.6) is Divergent for "p" between 0 and 1. Hence, by the Integral Test, the series from 1 to inf. of 1/n^p must also be Divergent on this interval. Sorry for the confusion, but we had covered this in class at a previous time in a section on integrals. Cheers and thanks for watching!! Professor Leonard
Hats off, sir. I've got to be frank, it's really noble of you to be helping out students like that. The explanation can't be any more clear. Lucky students, I swear.
Professor Leonard I want to strongly thank you for these calculus II lectures that you have posted. I'm currently taking Cal II online and the online videos that has been provided by my professor literally puts me to sleep. Your videos make Cal II bearable and you explain how to do the sections clearly. Without your videos I would have called it quits for my pre-engineering degree. You are a career saver. Keep on keeping on and making learning fun :).
I never get tired of learning math when Professor Leonard is the teacher. He explains everything that's going on in a way that anybody can understand. Such a great teacher!
Prof. Leonard, thank you for helping me achieve my dreams of becoming a mathematics major! I love raising my hands and nodding while watching your videos.
I am starting graduate school this fall to earn my master's degree in mathematics. I've spent the summer of 2022 reviewing the calculus sequence in preparation, and even though I took (and aced) all three classes at my community college, Professor Leonard's videos have been invaluable in filling in gaps and reminding me of stuff I forgot or didn't quite get a grip on before.
These videos have practically saved my life. Thank you so much for your willingness to freely share. I never took precalc and was in calc 1 more than 8 years ago before taking calc 2, your videos literally pulled my grade from a 62 to 87, and that's with time constraints. I am sure I would have made an easy A if I had had enough time to watch these videos more diligently. You are super amazing at this professorship thing :-)
Professor Leonard, You are amazing! I wish I had enough money to be a patron of your channel RIGHT NOW! You helped me get through Calc 1, and now you are helping me get through Calc 2. Just know that it is FREE channels like yours that will lessen the achievement gap.
I find it kind of ironic that all the ads while watching these videos are for other online tutoring services while I am currently watching the best math tutoring ever! Thank you so much Professor Leonard- you're videos are amazing.
Most of the university lectures are just waste of time. These videos are much better than those and we can watch these WHENEVER we want and AS MUCH AS we want.
This guy always manages to say "Are you listening?" the moment I stop listening. Needless to say, I am eternally grateful for these videos (and the people who post timelines in the comments).
MUCH RESPECT AND MANY THANKS TO U PROF.LEONARD ,,THE REAL MEANING OF HELPFULNESS .. HOPE THAT ALL PROFS HAVE A SUCH WAY IN EXPLAINING ,, YOUR STUDENTS ARE VERY LUCKY
Professor Leonard do you teach linear algebra and diff. Equations? If you do, please upload videos for those classes!! Im currently in cal. 2 and have watched every single one of your videos since cal. 1. Im afraid that I have come to depend to much on your videos and don't think I will do well without your help since all the math professors at my school are beyond horrible! Please upload them if you do! You will save many lives!! Also, thank you for all the help you have provided already. I made an A in my cal. 1 class and have like a 98 right now in cal. 2 and it's all because of your beast teaching skills. Thank you very much professor! I really mean it!
thank you thank you for so willingly sharing your classroom with everyone. you are such a generous person and amazing teacher. do not question your career choice, this is clearly what you have been called to do!
Your videos are so awesome!! Every time after my Calculus 2 class I come out confused and go straight home to watch one of your videos and I completely understand everything!! Has helped me a lot with my class!! Thanks a bunch!!
I am so mad at myself that I just found about Professor Leonard's videos at the end of the year. These have helped me understand series and sequences so much better than any other youtube videos or my university professor. Thank you so much!
This is a great video it is really really helpful. Professor Leonard is infinitely better than my college Calc 2 professor. I just have one request. In the comment of the video i would like to see a table of contents with times where different topics start. It would be easier to find the specific topic or concept that we may be looking for. Other than that I really enjoy the videos. It is very helpful. I really appreciate this.
around 2:46 Professor Leonard was talking about Monotonic Sequences when you can express a sequence in terms of a function to find if it's increasing or decreasing
Thanks so much for teaching me. I've watched so many of your lectures to get me to this level of calc 2 and its always been difficult for me to study and to do math well. I love all you youtubers who help ppl learn!
Professor Leonard, thank you for a fantastic explanation of the Integral and P-Series Test for Convergence/Divergence in Calculus Two. The P-Series Test is simple to follow and understand, however showing a function is decreasing to satisfy the Integral Test is problematic. Many Calculus textbooks does not show this important step in their Integral Test examples.
Thought i'd leave a random comment, i don't know if you check these things. You're videos have been a major help. I recently got out of the service and haven't done real math in quite a long time. So, like a dummy i went ahead and started back full throttle (only with today's world online). I'm taking Cal 2 over the summer and the video lectures for the course are between 10-30 minutes long and don't go into Nearly enough detail. I am FAILING like a boss, BUT watching your videos has already brought me up to a higher failing grade! That doesn't sound like a compliment, but it is. So thanks for being a kick butt professor! Send some genius vibes my way if you read this! To everyone else struggling with Calculus, i feel your pain and hit me up if you want or need a study group, because i sure as heck do!
Just a quick note at 58:03 -- the limit is indeed 0 but you can't use l'Hopital's. The numerator e^(1/n) goes to 1 and the denominator n^2 goes to infinity so you simply get 0.
23:53 when Patrick asked if since the function is continuous, can't you just write it as ln(2x-1) ALL THE TIME instead of the absolute value. This was the exact same question i had in my head
These videos are saving my ass. Thank you for sharing the big picture, the little details, for explaining EVERYTHING. These classes remind me why I started math and why I enjoy it.
WHEN YOU SEE THE LECTURE IS ONLY AN HOUR LONG...YES! GREAT CONTENT! A LIFESAVER! GOD BLESS YOU! (THUMBS UP IF YOU YOU STARTED WRITING NOTES IN ALL CAPS TOO)
Hey Professor Leonard I hope your doing well and enjoying your summer. I just wanted to know as a regular viewer of the great channel if you intend to teach linear algebra and or differential equations? Please let me know as you are great at explaining your courses.
Can anyone help me at around 19:10? Professor Leonard says that he "doesn't know" whether the limit of the integral is equal to the sum of the series, but doesn't the fact that it is PCD pretty much guarantee that it is not the sum? Is there a counter example to my hypothesis?
Oh God.he save my CasII course. I dont think i need to go to myclass any more,due to i.dont even what my professor say in the class. By the way Amazing video🎉🎉
I don't understand how at 13:45 , he asks if it's positive on 1 to infinity but if x were to be 1 then wouldn't that make the denominator 0 thus making the function undefined?
thank you for helping us i live in middle east and i study industrial engineer i study at Jordan uni for science and technology and our teachers are master student and they really suck hope u can visit us someday thanx for helping us
I hope these students know how blessed they are to have such an amazing professor. You don't see teachers like this anymore.
+❤ Gina ❤ hip
Beyond blessed. Prof Leonard is one of a kind
+ViaVanille
I have a physics teacher in Wave Mechanics who is a lot like Leonard - he has an interesting habit of spending the last few minutes right before a lecture begins on playing random live shows of heavy metal bands for some reason, probably just to entertain us while we wait for class to begin, and he also makes a lot of references to Star Wars in a lot of his problems (he has even dressed up as Han Solo in some exam problems and made himself a character in those problems, lol).
He also has a rather awesome attitude to his computer during the classes - whenever it refuses to cooperate he is like "what a piece of crap =P", or casually asks the students for help with it.
He is always in a great mood as well, and clearly enjoys his job.
It is teachers like him and Leonard who make lectures fun.
Yes, and he can also help people lifting heavy boxes and cars.
Big fax, wish professor Leonard was my calc 2 instructor
COVID-19 brothers using Prof Leonard to save their semester after their calc II class went online, where you at!??!
here lmao
HERE
here
I am here test on series tomorrow
Test on Wednesday
00:00:00 INFO: Integral Test Explanation
*) f(x) = f(n) = a_n
*) f(n) must be continuous, positive & decreasing
*) If integral converges, so will series
*) Does not always provide sum of series
00:10:36 INFO: Integral Test Example 1
*) Step 1: Identify f(x)
*) Step 2: Determine if it is positive, continuous & decreasing
*) Step 3: Solve improper integral
*) Step 4: Connect the dots: lim f(x) exist ∵ series is convergent
00:19:40 INFO: Integral Test Example 2
00:26:27 INFO: Integral Test Example 3
*) Proving f(x) = ln(x)/x is decreasing
*) The integral diverges ∵ series is divergent
00:41:08 INFO: P Series Explanation
*) P series will always diverge when P 1
00:47:20 INFO: P Series Summary
00:49:38 INFO: P Series Example 1
00:51:00 INFO: P Series Example 2
*) Root P
00:52:00 INFO: P Series Example 3
*) Negative P
00:57:45 INFO: Integral Test Example 4
*) Will divergence test work? No
*) Is it a geometric series? No
*) Is it a p series? No
*) Can you use Integral Test? Yes, f(x) is positive, continuous & decreasing
brilliant Note-taking skills
You sir are a real badger!
Not all heroes wear capes
didn't watch the video, just have read this and aced the quiz. thanks!
God knows how many good teacher qualities I saw in him like explaining so amazingly ,not assuming we'll know-it-all instead explaining every step , asking short questions for maintaining attention of students ,complimenting (like good ,...) for correct answers, being patient and calm,making jokes and light environment in class...I am blessed to watch this!!!!!!! THANK YOU SIR SOOOO SOOOOO MUCH.
This guy should be teaching at MIT
Raquel Gallo Hello, I didn't elaborate because this particular example was covered in section 7.6 at 43:15. Please check it out! Basically, the divergence test for the example you are referencing here shows divergence for any "p" = 0 or "p" < 0. For "p" = 1, we create a Harmonic Series (Divergent). For "p" between 0 and 1, we use the example of an improper integral to show that the function of 1/x^p (I believe I use x^n in 7.6) is Divergent for "p" between 0 and 1. Hence, by the Integral Test, the series from 1 to inf. of 1/n^p must also be Divergent on this interval. Sorry for the confusion, but we had covered this in class at a previous time in a section on integrals. Cheers and thanks for watching!!
Professor Leonard
+Professor Leonard AQUA FRESH
+Professor Leonard makes sense!
you are awesome keep doing your thing pal. and oh yea... does that makes sense to you ? xD
Professor in 38:50 I used integration by parts. I got ln(x^lnx)*1/2. Is that right?
You are more than awesome professor!
Hats off, sir. I've got to be frank, it's really noble of you to be helping out students like that. The explanation can't be any more clear. Lucky students, I swear.
Professor Leonard I want to strongly thank you for these calculus II lectures that you have posted. I'm currently taking Cal II online and the online videos that has been provided by my professor literally puts me to sleep. Your videos make Cal II bearable and you explain how to do the sections clearly. Without your videos I would have called it quits for my pre-engineering degree. You are a career saver.
Keep on keeping on and making learning fun :).
You have saved me in Calc 2. What an AMAZING professor.
Wow do I even need to go to college? EVERYTHING is explained by you. simply amazing.
Seriously. My math professor isn't bad but I feel I am learning at a much faster pace here.
@StraightupB34ST
Sure, however you CAN prepare yourself as much as possible so that you can go and kick ass at the college later. =P
I never get tired of learning math when Professor Leonard is the teacher. He explains everything that's going on in a way that anybody can understand. Such a great teacher!
Prof. Leonard, thank you for helping me achieve my dreams of becoming a mathematics major! I love raising my hands and nodding while watching your videos.
I am starting graduate school this fall to earn my master's degree in mathematics. I've spent the summer of 2022 reviewing the calculus sequence in preparation, and even though I took (and aced) all three classes at my community college, Professor Leonard's videos have been invaluable in filling in gaps and reminding me of stuff I forgot or didn't quite get a grip on before.
These videos have practically saved my life. Thank you so much for your willingness to freely share. I never took precalc and was in calc 1 more than 8 years ago before taking calc 2, your videos literally pulled my grade from a 62 to 87, and that's with time constraints. I am sure I would have made an easy A if I had had enough time to watch these videos more diligently. You are super amazing at this professorship thing :-)
Professor Leonard,
You are amazing! I wish I had enough money to be a patron of your channel RIGHT NOW! You helped me get through Calc 1, and now you are helping me get through Calc 2. Just know that it is FREE channels like yours that will lessen the achievement gap.
I find it kind of ironic that all the ads while watching these videos are for other online tutoring services while I am currently watching the best math tutoring ever! Thank you so much Professor Leonard- you're videos are amazing.
When the teacher sincerely teaches this is the result! You are awesome, Prof!
His explanation is crystal clear. I am now preparing for the final and thanks to him, i can make my life easier. Love this professor!!!!!!
26:06 She said COVID 21, she KNEW about covid happening in 2021. Actual future scientist here!
GOD BLESS YOU! I watched all of your calc 2 videos and it paid off big time! without you I would have been so lost! You and PatrickJMT are legends.
Thank you so much leo u have no idea how many ppl u r supporting rn thanks ❤😊
Most of the university lectures are just waste of time. These videos are much better than those and we can watch these WHENEVER we want and AS MUCH AS we want.
I just wanna thank you for your help. I've passed my calculus exam thanks to your explanations. :)
Half of my class comes to this guy to learn calculus over our own professor lol
How are you doing in calc so far?
mine too!!
This man is the reason I'm going into my Economics Ph.D.
Thank God for Professor Leonard and the work he does lol
Awesome, best of luck to you!
This guy always manages to say "Are you listening?" the moment I stop listening. Needless to say, I am eternally grateful for these videos (and the people who post timelines in the comments).
MUCH RESPECT AND MANY THANKS TO U PROF.LEONARD ,,THE REAL MEANING OF HELPFULNESS .. HOPE THAT ALL PROFS HAVE A SUCH WAY IN EXPLAINING ,, YOUR STUDENTS ARE VERY LUCKY
He is honestly the best calc teacher I have ever had!
Professor Leonard do you teach linear algebra and diff. Equations? If you do, please upload videos for those classes!! Im currently in cal. 2 and have watched every single one of your videos since cal. 1. Im afraid that I have come to depend to much on your videos and don't think I will do well without your help since all the math professors at my school are beyond horrible! Please upload them if you do! You will save many lives!!
Also, thank you for all the help you have provided already. I made an A in my cal. 1 class and have like a 98 right now in cal. 2 and it's all because of your beast teaching skills. Thank you very much professor! I really mean it!
he teaches diff equations now
Thanks alot..from Kenya...this sides we dont get good professors hence youve saved me big time
P-Series : 41:18
you're the bomb dot com
ByshStoleMyFish Lmao, you reminded me of my 8th grade teacher. He used to say that a lot.
Thank you so much for explaining calculus in a way we can all understand. For repeating and making concepts as clear as possible
42:20 lol my ears , The best teacher ever
Thank you for such a helpful video! Professor Leonard: The Man, the Myth, the Math-Teaching Legend
thank you thank you for so willingly sharing your classroom with everyone. you are such a generous person and amazing teacher. do not question your career choice, this is clearly what you have been called to do!
Thank you for re-awakening the math in me. Your presentation. You are simply a great teacher.
Just reviewing some material, and I must say, this guy is a better lecturer than my actual profs that I have to PAY tuition for
Your videos are so awesome!! Every time after my Calculus 2 class I come out confused and go straight home to watch one of your videos and I completely understand everything!! Has helped me a lot with my class!! Thanks a bunch!!
I rely on these lectures cause my online professor for Calculus is crap. You helped me understand to steps to solving problems so thank you.
I am so mad at myself that I just found about Professor Leonard's videos at the end of the year. These have helped me understand series and sequences so much better than any other youtube videos or my university professor. Thank you so much!
lol
This is a great video it is really really helpful. Professor Leonard is infinitely better than my college Calc 2 professor. I just have one request. In the comment of the video i would like to see a table of contents with times where different topics start. It would be easier to find the specific topic or concept that we may be looking for. Other than that I really enjoy the videos. It is very helpful. I really appreciate this.
I have a quiz tomorrow on my calculus2 and Professor Leonard is my hero
around 2:46 Professor Leonard was talking about Monotonic Sequences when you can express a sequence in terms of a function to find if it's increasing or decreasing
He is so amazing. This is how I learn when I have such a horrible calc 2 professor.
i lost confidence in maths but watching ur vidoes restored my hope .prof u are the best of the best .keep it up God bless u
I am from Simon Fraser University. Really thank you, Professor Leonard. I will let my friends to know you.
Thanks so much for teaching me. I've watched so many of your lectures to get me to this level of calc 2 and its always been difficult for me to study and to do math well. I love all you youtubers who help ppl learn!
Professor Leonard, thank you for a fantastic explanation of the Integral and P-Series Test for Convergence/Divergence in Calculus Two. The P-Series Test is simple to follow and understand, however showing a function is decreasing to satisfy the Integral Test is problematic. Many Calculus textbooks does not show this important step in their Integral Test examples.
at minute 41:00 aren't we supposed to substitute 3 and b in our U that we took to get our new bound or interval?
Thank you very much Sir. You have been of tremendous help to me.
@Professor Leonard This helped me a lot, you use the same method as my curriculum’s professor but you explain it in a much clearer way! Thank you
Thought i'd leave a random comment, i don't know if you check these things. You're videos have been a major help. I recently got out of the service and haven't done real math in quite a long time. So, like a dummy i went ahead and started back full throttle (only with today's world online). I'm taking Cal 2 over the summer and the video lectures for the course are between 10-30 minutes long and don't go into Nearly enough detail. I am FAILING like a boss, BUT watching your videos has already brought me up to a higher failing grade! That doesn't sound like a compliment, but it is. So thanks for being a kick butt professor! Send some genius vibes my way if you read this!
To everyone else struggling with Calculus, i feel your pain and hit me up if you want or need a study group, because i sure as heck do!
Great idea
Perfect videos for taking Cal II during a pandemic!
Am definitely going to graduate,these videos help a lot. THANKS PROF
Just a quick note at 58:03 -- the limit is indeed 0 but you can't use l'Hopital's. The numerator e^(1/n) goes to 1 and the denominator n^2 goes to infinity so you simply get 0.
Professor Leonard is the best. First he hits the gym, then he hits the books.
42:22
Then: bless you
2020: run 🏃🏻♂️
you are very amazing prof., your students are very blessed ,you real know your story,
This was more helpful then my uni prof! Thank you!:)
Thank you sir thanks🌹🌹🌹🌹🌹🌹🌹🙏🙏🙏🙏🙏🙏🙏
this helps a lot in integral test for series, appreciate for the video :)
Man i respect how he said i dont know there....very conscientious of him
What an amazing guy. Lucky kids.
23:53 when Patrick asked if since the function is continuous, can't you just write it as ln(2x-1) ALL THE TIME instead of the absolute value. This was the exact same question i had in my head
Thank you, I passed my final test because of you
you are simply making calculus our fovourite meal!!!!
These videos are saving my ass. Thank you for sharing the big picture, the little details, for explaining EVERYTHING. These classes remind me why I started math and why I enjoy it.
WHEN YOU SEE THE LECTURE IS ONLY AN HOUR LONG...YES! GREAT CONTENT! A LIFESAVER! GOD BLESS YOU! (THUMBS UP IF YOU YOU STARTED WRITING NOTES IN ALL CAPS TOO)
Thank you so much very helpful!
Wowww I thought I understood the integral test pretty well but 7 minutes in and I had a moment of clarity 😳 thank you
Some teachers can't even explain what the fuck an integral is
Then there's this god-given all-powerful omniscient prophet of mathematics.
Test tomorrow and I pulled a pro gamer move and studied all of the infinite series and sequences today ;)
The only guy that makes Cal 2 make sense 💜💚💗❤💞🔅🔅
30:01 dropped my pen when he said " don't take e "
"take e of both sides, but don't take E that's bad for you" loooool
Really great video.....you express concepts so distinctly, I wish my professor could emulate some of your qualities
Oh god I love your lessons... I can understand everything. Thank uuuuuuuuuu
you my sir, are a life saver :)
Hey Professor Leonard I hope your doing well and enjoying your summer. I just wanted to know as a regular viewer of the great channel if you intend to teach linear algebra and or differential equations? Please let me know as you are great at explaining your courses.
Currently watching a marathon of his videos to get ready for finals lol
Fantastic video...thanks for sharing.
sir
you are the best teacher....
Thank you from students in 2020
Integral Test feels like pulling out an AK47 to kill a deer
thanks Professor Leonard
Such an amazing professor
Can anyone help me at around 19:10? Professor Leonard says that he "doesn't know" whether the limit of the integral is equal to the sum of the series, but doesn't the fact that it is PCD pretty much guarantee that it is not the sum? Is there a counter example to my hypothesis?
Oh God.he save my CasII course. I dont think i need to go to myclass any more,due to i.dont even what my professor say in the class. By the way Amazing video🎉🎉
a thumbs up from India
Thank you, Professor Leonard !
Thank you soo much , you have helped me a lot. Thank you sooo much
thank you professor I am from somalia
I don't understand how at 13:45 , he asks if it's positive on 1 to infinity but if x were to be 1 then wouldn't that make the denominator 0 thus making the function undefined?
thanks a lot this was very helpful
thank a lot professoe leanord. i love calculus
thank you for helping us i live in middle east and i study industrial engineer i study at Jordan uni for science and technology and our teachers are master student and they really suck hope u can visit us someday thanx for helping us
By the way, the requirements for the function can be relaxed to non-increasing and non-negative instead of continuous, decreasing and positive.
you are the best Sir!!!
Simply love this.
I wish I could give Prof Leonard my tuition payments instead