Differentiating and Integrating Power Series

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  • เผยแพร่เมื่อ 23 ธ.ค. 2024

ความคิดเห็น • 375

  • @maxkline8985
    @maxkline8985 10 ปีที่แล้ว +553

    ive decided im just going to get this wrong on the test.

    • @musamoloi2149
      @musamoloi2149 10 ปีที่แล้ว +14

      lol you and me both...

    • @ifoundthewords
      @ifoundthewords 10 ปีที่แล้ว

      Lol :P

    • @AnotherMatt
      @AnotherMatt 9 ปีที่แล้ว +1

      +Max Kline I'm thinking the same thing

    • @Brownkevin7
      @Brownkevin7 8 ปีที่แล้ว

      join the club

    • @REALSLIK
      @REALSLIK 7 ปีที่แล้ว +2

      Max Kline tried that last summer..got an F

  • @TheFunkyCriminal
    @TheFunkyCriminal 9 ปีที่แล้ว +67

    you are doing the lord's work patrick

  • @spencershrek1491
    @spencershrek1491 8 ปีที่แล้ว +15

    from me, at 2am before my 9am exam, bless your soul. BLESS YOUR SOUL, YOU BEAUTIFUL MAN

  • @Hdhillon90
    @Hdhillon90 12 ปีที่แล้ว +2

    I really hope you realize how much your help is worth. This type of help from anybody costs tremendous amounts of money and sometimes you dont get what you pay for. You are by far one of the best tutors i have ever came across. I wish there where more people like you so getting help was easier. But you are definitely a man with a good heart and intentions. Keep up the great work patrick, you're the man!

  • @lanfordvideos
    @lanfordvideos 6 ปีที่แล้ว +47

    I am choosing to believe in faith that I will actually be able to understand this concept by my tenth time viewing this video.

  • @Sublime_37
    @Sublime_37 6 ปีที่แล้ว +2

    I loathe math and most math professors. This guy is the definition of a life saver, by far!

  • @sammybarajas1997
    @sammybarajas1997 9 ปีที่แล้ว +130

    This made me want to cry.

  • @janitarjanitar
    @janitarjanitar 14 ปีที่แล้ว +2

    this is the first topic in calc so far really tripped me up. thanks for all the time you spent on this chapter

  • @steelblue831
    @steelblue831 14 ปีที่แล้ว +2

    Patrick! What can I say, man, except you rock! I am taking an online calc II course with very little contact with the instructor but need to the course to get into my grad school that starts in sept. I would, without a doubt, be miserably failing this course without all your videos. You shine light in dark places, bro. Much appreciated!

  • @richardreber3878
    @richardreber3878 3 ปีที่แล้ว

    I am a physician by profession who just loves studying this stuff for enjoyment. PatrickJMT your videos are wonderful and so helpful. Keep up the good work.

  • @mytwohands
    @mytwohands 13 ปีที่แล้ว +3

    @patrickJMT I've been watching Khan Academy for the last few months. You really picked up where he left off in the Calculus series. I'm totally a Patrick JMT convert and tell everyone in my calc class about it. I really think if you were to collaborate, it would take the educational community by storm. Thank you for all you do here! I will definitely donate to the cause as soon as I'm not a terribly poor college student!

  • @TheGuyWithNoVideos12
    @TheGuyWithNoVideos12 4 ปีที่แล้ว +5

    This is probably the best video on this topic I have ever seen. I've been through Khan Academy and some other random videos including the ones that come with my online homework and this really helped me understand. Especially during a time where I can't get in person help (thanks COVID-19)

    • @patrickjmt
      @patrickjmt  4 ปีที่แล้ว +1

      thanks! spread the word :)

  • @lukewooly6104
    @lukewooly6104 4 ปีที่แล้ว

    This guy explained a week's worth of lectures in about 10 minutes thank you

  • @CoconutZai
    @CoconutZai 12 ปีที่แล้ว

    Thanks for helping to promote the love of math and helping me understand more about differentiating and integrating a power series in just 10 minutes.

  • @patrickjmt
    @patrickjmt  14 ปีที่แล้ว

    @Jawshooah it is a function of x.

  • @andrewd1763
    @andrewd1763 10 ปีที่แล้ว +31

    even the patrick, this stuff hurts my brain.

  • @peslie25
    @peslie25 14 ปีที่แล้ว +1

    i've been watching all of your calculus videos since my midterm, since then my grade has gone up TWO LETTER GRADES. Tomorrow morning is my calculus 2 final and if i make an A on it i make an A on the course. i dont wanna jinx myself but ive been studying like crazy with the help of your videos so i feel good. If i get an A i will donate my life savings to you and name my first child patrick. no lies.

    • @redpotter4074
      @redpotter4074 2 ปีที่แล้ว

      11 years later but how did you do?

  • @sophiaoku4020
    @sophiaoku4020 9 ปีที่แล้ว +157

    this math topic has ruined my LIFE

  • @karama8484
    @karama8484 15 ปีที่แล้ว

    Man patrickJMT... the way you simplified that last problem... I would've never thought of that...thx a lot for your vids..

  • @akilaes777
    @akilaes777 8 ปีที่แล้ว +77

    when its 2:30 am and you have an exam the next morning

    • @rosalo2096
      @rosalo2096 8 ปีที่แล้ว +16

      TheAvocado get at my level 5 am

    • @themagickalmagickman
      @themagickalmagickman 3 ปีที่แล้ว +1

      same story 5 years later

    • @akilaes777
      @akilaes777 3 ปีที่แล้ว +1

      @@themagickalmagickman wow 😂 glad I can say I’ve graduated and have a job now

  • @patrickjmt
    @patrickjmt  15 ปีที่แล้ว

    yes, you are correct that i should have emphasized this.
    i was focused on the mechanics when i made the problem.
    i also made this video for a student that i was tutoring at the time, and she was aware of the interval, so i did not think so much about it : )

  • @gemvampgrl18
    @gemvampgrl18 16 ปีที่แล้ว

    I love you. You just made my 2am study session. I wish all teachers and text books used the more tricky examples instead of annoyingly simple, non-relevant things like 1/(1+x). *hugs* you ROCK

  • @patrickjmt
    @patrickjmt  14 ปีที่แล้ว

    @iYoungJ89 yes, i believe it was my sophomore year (actually, it may have been the summer between freshman and sophomore year). but i came to math relatively late in life i think

  • @metalhead33
    @metalhead33 11 ปีที่แล้ว

    You deserve the nobel piece prize for changing so many lives!!!

  • @ChanakaJ
    @ChanakaJ 9 ปีที่แล้ว +70

    ooooohhhhhhhhhh
    Who's cathrina?? :)

  • @iamloveandpeace
    @iamloveandpeace 9 ปีที่แล้ว +53

    I am so lost

  • @erikpotenza1012
    @erikpotenza1012 10 ปีที่แล้ว

    Thank you so much man. Your explanation was clear and to the point, yet not taking shortcuts. I am still working on some of them, and still getting the wrong answer every now and then however my skills in diff and int power series has improved a ton thanks to your video.

  • @VikashKhandelwalworld
    @VikashKhandelwalworld 12 ปีที่แล้ว

    Best video everything told in the most wisest and quickest way.May be not attanding long classes and seeing 10 mins video are enough.

  • @patrickjmt
    @patrickjmt  16 ปีที่แล้ว

    The key observation part, why did you use(1/1-2x) instead of the original [1/(1-2x)^2] ---> cause we know a power series representation for 1/(1-2x)
    on the integrating and derivative part... i am just using the normal rules take one away from the power or add one to the power

  • @FlashFire40
    @FlashFire40 12 ปีที่แล้ว

    Ah thank god this video exists. This was the only thing in what we covered in sequences and series where I was really confused. Thanks a ton!

  • @sharjah12345
    @sharjah12345 15 ปีที่แล้ว

    yo patrickJMT, that was indeed a gr8 help for my finals, thnx a ton man. I needed this help bcz I missed the class for this lecture and this tutorial saved my time , thnx a ton once again.

  • @ipons-jv9ll
    @ipons-jv9ll 7 ปีที่แล้ว

    Thanks a ton for your videos. They've been invaluable in the places my textbook and professor didn't cover thoroughly, especially this one.

  • @patrickjmt
    @patrickjmt  15 ปีที่แล้ว

    well, you can show it diverges through an arithmetic argument; to show that it diverges 'slowly'... what does 'slowly' mean? i agree.... if you add up terms... lots and lots and lots and lots of them, you still getting a pretty small number

  • @NotLegato
    @NotLegato 7 ปีที่แล้ว +43

    oh jesus christ why does this exist.

  • @chopinandliszt
    @chopinandliszt 15 ปีที่แล้ว

    The most enthusiastic teacher I've ever seen. You turned off your phone, which rang during the seminar to keep it going. Even my passionate teacher doesn't do that. PATRICK JMT ROX!!!!!!!!!
    Anyway, how do we show that S(1/n) (summation of 1/n as n goes from 1 to infinity) diverges extremely slowly? We can't do it using Test for Divergence.

  • @austincb90
    @austincb90 8 ปีที่แล้ว +7

    At 2:47, why does the index now start at 1?

    • @Yhozen
      @Yhozen 7 ปีที่แล้ว +1

      Because at n = 0 you will start by adding 0, so you can skip n = 0

    • @marcbenedict3676
      @marcbenedict3676 6 ปีที่แล้ว

      from what i understand, whenever you take the derivative of the summation formula you add 1 to n

    • @DarkKinghtrulz
      @DarkKinghtrulz 6 ปีที่แล้ว +1

      Well the first series starts with a constant. In it's derivative, that constant from the original series will be zero because the derivative of a constant is zero. So the next term in the series will be a constant. By increasing the index to one, it accounts for that.

  • @greta7885
    @greta7885 8 ปีที่แล้ว +1

    This is such a hard topic and I couldn't follow my professor or TA . Thanks for breaking it down into smaller steps

  • @TheMagnas26
    @TheMagnas26 12 ปีที่แล้ว

    I can't thank you enough for all the helpful videos you have made. It might be the way of your speaking, the explanation, but let be clear, you are a much better professor than my lecturer in my university! Every time I see one of your video, I feel more confident and appreciate more about math. Why the hell in my university (despite its reputation) have only crap teacher ?
    Thanks again, you are the best !

  • @jrggarcia1
    @jrggarcia1 4 ปีที่แล้ว

    Why at 7:15 did not add 1 to the (-4)^n and add it everywhere else after changing the index. On the first problem you added it to the (2^n)

  • @RRR66620
    @RRR66620 9 ปีที่แล้ว +5

    Why didn't you change the index after integration?

  • @Link64
    @Link64 14 ปีที่แล้ว

    ommmmgggg wtf was that?!?!?! All that was just like.....woooaaaaahhhhhh, so crazy. I didn't understand what my professor was talking about when he did this lecture this morning. But after watching this, I actually kind of understand it now : D. I just have to watch this a couple of more times to fully understand everything. Thank you for making everything much simpler to understand!

  • @patrickjmt
    @patrickjmt  15 ปีที่แล้ว

    yes, you are correct, i should have emphasized this.
    when i made the video, i was focused on the mechanics of the problem

  • @curtistrinh5654
    @curtistrinh5654 7 หลายเดือนก่อน

    Is there a paritcular reason why you dropped dx in the moment 6:59 of the video? I'm curious.. is it because instead of the sum it becomes an integral or is just a semantic error? when do we drop dx? when we integrate/differentiate? and do we keep it with the summation symbol sigma? exam is on fri 8:00 AM.

  • @patrickjmt
    @patrickjmt  13 ปีที่แล้ว

    @abthurd you should watch the video about: radius of convergence

  • @DieNo0b
    @DieNo0b 14 ปีที่แล้ว +1

    Hi thanks for the vid. A question though. Why dont you shift the index by replacing n with n-1 in integration, much like how you replace n by n+1 in differentiation?
    Thanks in advance

  • @DRsFINESTCR123
    @DRsFINESTCR123 10 ปีที่แล้ว

    At 2:53, you state that you get a constant out which means you need to start the series at 1, I don't really understand why. What exactly is going on there?

  • @haha-yn6es
    @haha-yn6es 8 ปีที่แล้ว

    The power of series for tan(2x) is valid between -1/4 and 1/4. At 8:20, Patrick plug in x as 0, in order to solve C. Theoretically, can he put in other numbers, such as 1/8 or -1/16?

    • @christianrodriguez823
      @christianrodriguez823 8 ปีที่แล้ว

      haha Sure, but I don't see why you'd want to since plugging in 0 gives you such a nice answer right off the bat. I'm sure you'll still get that C=0 for any number within the interval of convergence.

  • @Duhhh233
    @Duhhh233 13 ปีที่แล้ว

    You are awesome! I was just about to give up on my homework, but I think I get it now. Thanks for putting this up!

  • @eracela8341
    @eracela8341 11 ปีที่แล้ว

    n is like a constant in the case of summations. For example, summation from 1 to 3 (2^n)(x^n) = 2x + (2^2)(x^2) + (2^3)(x^3). When you differentiate this you end up with 2 + (2^2)(2x) + (2^3)(3x^2). So if you represent it in summations, it is summation from 1 to 3 (2^n)(n)[x^(n-1)].

  • @secondo92
    @secondo92 8 ปีที่แล้ว +1

    Although the phone ringing made my ADHD explode, you explained it so well that it was very easy to understand! Thank you!

  • @GohansRiceBall
    @GohansRiceBall 13 ปีที่แล้ว

    Thanks for the video. That was pretty cool seeing how you could manipulate a function to get the power series representation. It probably does take more examples to get used to it, and I'm all for more seeing more examples.

  • @syntheticpolymer620
    @syntheticpolymer620 4 ปีที่แล้ว

    great vid! I couldn't find a vid that had exactly what I needed until this!

  • @cristianrodriguez1782
    @cristianrodriguez1782 8 ปีที่แล้ว

    The level of joy I'm feeling right now before my test

  • @patrickjmt
    @patrickjmt  15 ปีที่แล้ว

    i have no clue why those are there... (or were therE, cause i just removed them).
    thanks!

  • @zoom2121
    @zoom2121 14 ปีที่แล้ว

    Please do more examples, this is pretty next level compared to the other stuff

  • @Cramerica66
    @Cramerica66 11 ปีที่แล้ว

    This has to be the hardest thing to wrap my mind around so far, originally it was the disc and shell method but this takes the cake. I suppose that is how most new topics of math feel.

  • @FashionZombies27
    @FashionZombies27 12 ปีที่แล้ว +1

    I'm curious as to what happens at 5:37... How did the 2^n+1 multiplied by the x^2/2 become 2^n? Sorry! Been trying to figure out and for some reason, my brains not working right now.

  • @Jaxis96
    @Jaxis96 16 ปีที่แล้ว

    Hey thanks for posting these videos since im so lost on this topic but i had a few questions that I did not understand in ur examples
    Ex1: The key observation part, why did you use(1/1-2x) instead of the original [1/(1-2x)^2]
    Also, could u clarify how u found the derivative of 2^n x^n, i wasnt too sure how you got x^n-1 and also how did you determine what n to start off with (ie change from 0 to 1)
    Ex2: how u did the integration (-4)^n x^2n and why u ended up dividing by 2n+1
    thx

  • @MrMrannoying
    @MrMrannoying 6 ปีที่แล้ว

    Wouldnt the 2 near the very end that was outside the series distribute to both the 2^2n AND the x^2n+1? Also why did you use x=0 again for the last problem? Is that the only spot where this converges on?

  • @KakinTseGamePlayer
    @KakinTseGamePlayer 7 ปีที่แล้ว +9

    "Weeellllllll... Theeeerrrrreeeeesssss... myyy-"
    *Changes to 2x speed*
    "shamelessplugalrightsointhisvideowetalkaboutdifferenciatingandintegratingpowerseries..."

  • @patrickjmt
    @patrickjmt  15 ปีที่แล้ว

    well, it depends.
    expand out the series expansion, differentiate, and put it back together and you will see

  • @whoru206
    @whoru206 14 ปีที่แล้ว

    Patrick! Could you explain more why you make the n=1 at 2:39?
    I still don't understand why you had to change it from n=0 to n=1.
    THX!

  • @jasonjkeller79
    @jasonjkeller79 13 ปีที่แล้ว

    Excellent videos man! Like many others I'm stuck teaching myself because my actual professor, and I use that term loosely, is useless. Thanks to you, and others like you, I'm actually making sense of this stuff. So thanks again, and if possible I wouldn't mind seeing some more examples on this subject. Cheers!

  • @Blaisem
    @Blaisem 12 ปีที่แล้ว

    Do you choose the approach of deriving the power series via a differential equation or integration based on whichever method is easier to derivate/integrate?
    Thanks for this helpful video. I've got a tough physics question and haven't had much luck looking elsewhere, but this brings me a step closer to solving the problem. Again I appreciate your time for making this video.

  • @floyd617
    @floyd617 13 ปีที่แล้ว

    Hey, for the second question you did; what if you got c to equal a number.. then if you subracted it from the equation you got to make it look like the original question it wouldn't be a power series anymore right? so what would you do? and btw thanks so much patrick for providing us with so much help.. it means a lot to me and everyone else.

  • @patrickjmt
    @patrickjmt  13 ปีที่แล้ว

    @scilabo ok, go tell them! not much i can do about it

  • @TheXsnipexx
    @TheXsnipexx 13 ปีที่แล้ว

    @patrickJMT I am studying to become a math instructor. What would make calculus better in a classroom? Although I have an excellent professor, I do most of my learning on my own and with the great help of your videos.

  • @carlampp68
    @carlampp68 14 ปีที่แล้ว

    This video is the best! Thanks so much for all your help!

  • @ArazSalahli
    @ArazSalahli 12 ปีที่แล้ว

    thank you very much it helped me to understand the Differentiating and Integrating Power Series after 4 hours:)

  • @JC-wg5xn
    @JC-wg5xn 3 ปีที่แล้ว

    1:44 did I just hear a landline?? That takes me back.

  • @pinkpaulina
    @pinkpaulina 11 ปีที่แล้ว

    what about the integral x^2*ln(1+x) dx ?? I have been trying to do this one but I keep getting Sum(n=1 to inf) (-1)^n-1 * x^(n+3)/((n)(n+3)) +c... I don't understand what I am doing wrong!! help me please:)!!!! the answer in the back of the book is Sum(n=1 to inf) (-1)^n* x^(n+3)/((n)(n+3)) +c

  • @hanjis5894
    @hanjis5894 2 ปีที่แล้ว

    dang I was four years old when this video was uploaded and here I am using it for class

  • @bahstanchyeah
    @bahstanchyeah 15 ปีที่แล้ว

    for the problem in which you differentiated 1/(1-2x), how would you find the radius of convergence of that? Is it just one? I'm doing my math homework now and I had a very similar problem however, it was telling me that the radius of convergence was not one.

  • @toushi100
    @toushi100 8 ปีที่แล้ว +98

    i dont wanna study... i wanna play fallout 4

    • @patrickjmt
      @patrickjmt  8 ปีที่แล้ว +17

      +AHMED SALAH (TOUSHI) so go play

    • @toushi100
      @toushi100 8 ปีที่แล้ว +5

      +patrickJMT just finished my exam and thank to you i did really good :D

    • @CmG626xX
      @CmG626xX 8 ปีที่แล้ว +1

      omg wtf i scrolled down thinking exactly that! xD

    • @AnonYmous-pl2id
      @AnonYmous-pl2id 8 ปีที่แล้ว +1

      This is weird. I just reinstalled FO4 so I could play it when I finish studying.

    • @toushi100
      @toushi100 8 ปีที่แล้ว +1

      you know whats more weird?? i just started my midterms and i have an exam in like three hours and i dont wanna study i wanna play skyrim remastered

  • @davidnussbaum8092
    @davidnussbaum8092 6 ปีที่แล้ว

    Great video! I was really lost on my HW untill I watched this.

  • @patrickjmt
    @patrickjmt  15 ปีที่แล้ว

    glad that i have been able to help you : )

  • @graceface000
    @graceface000 10 ปีที่แล้ว

    Why did you decrease the index? Is it necessary?

  • @haneenwalid1652
    @haneenwalid1652 7 ปีที่แล้ว

    Thank you man you saved my whole degree

  • @rayhandler1673
    @rayhandler1673 8 ปีที่แล้ว

    in the final answer, dont you think that the index should be -2 coz you inserted the x^2? im confused...

  • @ccase2k
    @ccase2k 11 ปีที่แล้ว

    "there's my shameless plug" hahaha you're the man Pat! No shame in what you do brother, you've helped me through calc I and II, and I'll be damned if I don't donate to you after I graduate!

  • @sark96
    @sark96 9 ปีที่แล้ว +5

    this saved my life

    • @lifeoftomi_
      @lifeoftomi_ 9 ปีที่แล้ว +4

      Sarah Khan you should have gone to khan academy ;p

    • @SuperMoreno18
      @SuperMoreno18 9 ปีที่แล้ว

      +Seeker ooo

  • @thealfiggy
    @thealfiggy 12 ปีที่แล้ว

    Thank you so much for all of your videos, seriously you're helping me so much!

  • @BenJaminLongTime
    @BenJaminLongTime 9 ปีที่แล้ว

    so if i wanted the radius of this which version of the original function do I pull it from? f(x), d/dx, or the integrated one? Probably obvious if I knew what was going on with all this but I don't...

  • @lennyadams8275
    @lennyadams8275 7 ปีที่แล้ว

    These are such great examples! Thanks, Pat!! :D

  • @StarVans
    @StarVans 13 ปีที่แล้ว

    Just wondering... do you know when to change the n values? like n=0? n=1? when to change it? because I have no idea when to change it..

  • @SGvince22
    @SGvince22 13 ปีที่แล้ว

    i see what you did at 3:00, you differentiated the summation itself. what we were taught to do was, write out a few terms of the series, differentiate each, and find a pattern so you can write it in general form.

  • @ronijun888
    @ronijun888 14 ปีที่แล้ว

    do you have a lesson about the matrix inverse? The Gauss-Jordan Method?

  • @patrickjmt
    @patrickjmt  14 ปีที่แล้ว

    @curbsidelexi no prob, keep up the good work

  • @DaRaginGamer
    @DaRaginGamer 4 ปีที่แล้ว

    Why -4x^2 at 6:35?

  • @Daiz7788
    @Daiz7788 6 ปีที่แล้ว

    I'm still confused. What happened to the 2 over the (1-2x)^2? Also do you have to shift the index?

  • @Ari-jm6xx
    @Ari-jm6xx 5 ปีที่แล้ว

    When do you integrate and when do you differentiate?

  • @unrealriot
    @unrealriot 15 ปีที่แล้ว

    can you do examples of finding the radius of convergence after doing differentiating and integrating power series? Thanks.

  • @pitelis
    @pitelis 15 ปีที่แล้ว

    the proof of differentiation of power series and Abel's Theorem are very interesting too.

  • @JDMaxton1999
    @JDMaxton1999 4 ปีที่แล้ว

    Does differentiation change the radius of convergence?

  • @Blaisem
    @Blaisem 12 ปีที่แล้ว

    Why did he take the approach of deriving the series from 1/(1-2x)^2, then multiplying it by x^2/2? Is this preferable to instead deriving the series from x^2 first, then multiplying by 1/(1-2x)^2? Thanks.

  • @benjaminedwards7852
    @benjaminedwards7852 7 ปีที่แล้ว

    Thanks so much, this video and your other series helped clarify this area of calc 2. Wish my professor could explain it like this.

    • @joshurlay
      @joshurlay 7 ปีที่แล้ว +1

      Benjamin Edwards My test is tomorrow. Good luck on your subject areas too. My teacher already explained it really well and gave us ample notes on the topic but I'm on TH-cam trying to find something to give me an edge.

  • @concisestem
    @concisestem 13 ปีที่แล้ว +2

    Oh my goodness! I started laughing so hard when I heard the phone start to ring and even harder when you ran to silence it. Those footsteps were priceless. By the way, did I give you that t-shirt idea or were you gonna do it anyway?

  • @jewtuber08
    @jewtuber08 12 ปีที่แล้ว

    what does changing the index do to the n value in the series? why do you change the index to n=1 and then take one n away from the series.. if that makes any sense

  • @oneinabillion654
    @oneinabillion654 5 ปีที่แล้ว

    This is third day and I reached this video already. I started from first video of Cal 2 playlist. Just finished A levels yesterday

  • @WediWelWel
    @WediWelWel 11 ปีที่แล้ว

    I have your channel page bookmarked, that's how much your videos mean to me