Taylor series video!

Outstanding! A video connecting linear algebra with bases to polynomials would be zen. Taylor series and Fourier series are kissing cousins. They have different bases. I love the patreon only content.

Good suggestions. I may, if it looks like there's demand for it, add a video about Lagrange error bounds sometime down the road, so that might be a good chance to really dig into how good/bad the error is at various distances.

3blue1brown

Good point, I'll add it.

3blue1brown

Thanks so much, I really appreciate it.

3blue1brown

Wonderful series the whole calculus thing, made me become a Patron for these !

One little point: at 5:36 you mention some "exact" value of cos (0.1) but only give a finite fraction. Three dots maybe in order unless that finite fraction is really the exact function value at 0.1

Lionel Pöffel

For this you could display the absolute value of the deviation between the actual function and its finite taylor approximation.

Lionel Pöffel

Great video. You may want to illustrate the fact that far away from the sweet spot at which we know all derivatives any finite part of the taylor series could get arbitrarily bad.z

Lionel Pöffel

Another outstanding video, Grant. Thank you very much.

Jason Paul DeMont

Great video, thanks a lot! Is it possible to have a video showing why/how higher order derivatives can or cannot propagate far from a certain point on a function?

Great video! Thanks :-) I haven't quite put my finger on it yet, but the ultimate solution(s) to how our universe actually ticks will entail series like these. You're helping me visualize it :-)

Karin Rodrigues

I would love to! There's so much I want to do...

3blue1brown

Thank you for this video! It is one of the best and most well done so far. Still, I did not understand why when you make a Taylor series at a point, other than 0, you put in (x-a) in the Taylor polynomial. Also, at 15:50, after the circle that points to the triangle disappears, it is left a small curve on the rectangle beneath it, where the circle used to be. It disappears too after a few seconds, so I don't know if you care about that, it is a pretty small detail.

I think a lot of people would appreciate it if you could go into multivar as well at some point, and help explain things like curl, divergence, Greene's theorem, integrating 3d shapes, etc. At least for me my college course on multi was pretty rushed so I never really appreciated those ideas until I used them in later classes, and if you could do as good a job visually explaining those things as you did with your linear algebra series it would be a godsend to many struggling students.

You were so close to deriving Euler's formula. I know you do it in other videos. Maybe just hint at this line of reasoning for interested viewers to find the relationship themselves. Or direct them to your other videos.

Duncan Fairbanks

Interesting thought. At the moment, I think the video might be long enough as it is, but perhaps its worth adding a little note/exercise to this effect.

3blue1brown

Thanks so much, I really hope it can help anyone else with a similar calc II experience.

3blue1brown

Same typo "piont" at 21:45 just in case your fix above didn't fix it in both places.

Steve Muench

The ability to generate an entire function from any single point (for many functions) is similar to how any cell of the body contains the genetic instructions for the entire body.

Magnasium

I had a very rocky calc II experience that was focused only on computation, so I've never really had the pleasure of understanding why Taylor series work... until today. Thank you for that! And thanks for this series as a whole - it really is excellent. I'm already brainstorming ways I can use some of these videos when I teach calc I again in the fall and getting excited about some of the possibilities your videos allow. Keep up all the good work.

Suggestion: I thought that it might be worthwhile adding a small example on the trivial example of the Taylor Polynomial of polynomials... sort of like to close the loop. Of course the answer will be itself, but the process seemed to help me with the direct connection.

Christopher Burke

This is the greatest math video ever made.

Thanks so much Jacob!

3blue1brown

Thanks so much, good catch!

3blue1brown

Wondering - if your series of videos is really a series then it will go on for ever. Here is hoping :)

Christopher Burke

Have you already noticed and corrected the rather ugly typo around 10.40 ? Very nice series, I ought to add.

Oh, figured it out. If you open the video on YouTube there is a playlist.

Just wanted to say thanks for this series^^. You and Khan are the primary reasons to why I don't get anything out of calculus class^^.

Hi, thank you for the videos, I am new to the site. Is there a playlist of all chapters that I could watch? Its not very organized on patreon. Thank you and have a nice day

This was one of the best videos you've done so far! Really awesome explanation. I learned a lot about them at university but this video definitely made me really understand how and why these work - especially the part why there are "always" factorials!

Fantastic video! The animation for repeated iterations of the power rule was SO SATISFYING

Connor Alexander McCranie

Grant.... that was phenomenal. One of my favorite videos of yours. Congrats on rolling out the series after all these months. I've already decided, my Calc III students this summer will be able to get bonus credit by answering a question at the end of their quizzes based on the main ideas of your calc series, which is coming out just in time! So I've got about 25 views per video coming your way :)

Jacob Mirra

Hey Grant - I'm in the middle of watching it - great work again. I think there's a bit of a confusion at around the 12 minute mark, where you generalize finding the Taylor series for any function. You say that you plug in 0 to the function, but this is only for a Maclaurin, so I'd make that explicitly clear. You do that around 13 minutes, but I don't think that's clear until later.

Josh B.

Hey man, so, I'm torn with this comment. I haven't watched the video yet, but I'm kinda saving them to watch in order. I'm a new patreon supporter so I've only seen the first vid and the last one. Anyway, I have something to say about Taylor series. You may have mentioned it in the video, but I'm not going to watch it until I see all the others in order. After all, you'd never watch the last episode of a great show without watching the other episodes first. So here's the thing. Here's the question that I think is so important,or at least beautiful, for Taylor series. "If you knew EVERYTHING about a function at a single point x=a, how much of the rest of the function do you think you could predict? How far away from "a" could you go and still get an accurate prediction?" I think, for analytic functions, it's really beautiful that you can predict EVERYTHING. I found this really counter intuitive the first time I saw it. I can really predict f(x) if x is a billion units away from "a", simply based on knowing information at the point "a"? So yeah. I hope something like that made the cut for your video. I'll find out when I see it :) I teach first year calc and algebra for a "crash course" company in Vancouver. I will be suggesting your videos as MANDATORY for all my students. By far and away these are the best math videos I have ever seen. You are going to be a big part of a generation of mathematicians who actually understand wtf they are doing. Shel

Shel Hammer