Taylor series | Chapter 11, Essence of calculus

Taylor polynomials are incredibly powerful for approximations and analysis.
An equally valuable form of support is to simply share some of the videos.

Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced.

Timestamps
0:00 - Approximating cos(x)
8:24 - Generalizing
13:34 - e^x
14:25 - Geometric meaning of the second term
17:13 - Convergence issues

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: ngvutuan2811

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

Various social media stuffs:

[chastgibson]

WOW! I'm a calculus teacher. I have watched hundreds of hours of calculus videos always looking for ways to improve my own methods of explanation. This is by far the best math video I have ever seen. I am in awe. It literally gave me goose bumps.

[gobbedy]

Omg. This has to be one of the most brilliant math videos I've ever seen. Not just beautifully explained, but with amazing moving graphs, perfect syncing between explanations and animations, perfect rate of explanation, perfect tone. I'm just sitting here in awe. So thankful. SO thankful!!

[michaeladdis]

My Calc professor called them "tailored polynomials" in the sense that they are tailored to fit a function at a desired point

[ParadoxPython]

Taylor Series are one of the things I just could not grasp in my uni calculus class because of how dry and abstract everything was. I understand abstraction is important, but it helps so, so much to be led towards it from concrete examples rather than being thrown into its cold rapids right away. Thank you so much for closing this gap for me, you are a gift to humanity.

[BlackwaterPark]

Can we have a video where we just watch 3b1b animations of approximating functions with Taylor polynomials? That's so satisfying.

[iandavidson]

Finally understanding a new math concept is a spiritual experience.

[bruhnling]

Math teacher used this in class today instead of teaching it herself cause this video is THAT good, the teacher put aside her pride in favor of the amazing visuals. This is by far my favorite math channel and I was internally freaking out when she started playing it and I realized it was you. Probably the highlight of that class tbh

[AdhiNarayananYR]

Shout out to my math teachers at school and jee coaching centre who just wrote the formula for the Taylor series and proceeded to solve some example problems that may or may not appear in jee exam. And that was the end of it. All this time I was looking at this series as an ugly series until I watched this video. Under the guidance of the right teacher even the most mundane things do become beautiful. Thank you grant Sanderson for making these videos! Love from India 🇮🇳

[joeyaintwaffling]

Just thinking how mathematicians used to think all these, we need these extraordinary animation to just pick up the superficial part of it, truly they were marvelous.

[gpd]

Our math teacher speaks highly of your work and encourages us to watch your videos to learn more about the chapters we're working on.

He's definitely right, congrats sir.

Cheers from France :)

[davidmichels]

I am a calc 1 teacher for engineers and you just keep giving me amazing input to improve my lessons. Thank you!

[rewtru]

I studied Taylor polynomial expansion almost 10 years ago. I remember seeing the professor write the factorial at the denominator and wondering "What does the factorial come out of?" and also "Why isn't the reason why it does part of the class?" "Why isn't it explained explicitly on my book?" And finally I see this video. I looked it up it because I was sure you were going to reveal this to me. THANK YOU GRANT

[eriksundell]

@3Blue1Brown - I'm currently teaching students aged 16-17 about derivatives and integrals... The educational impact you make is immense! Please keep creating series about math! You have great narratives conveying beautiful insights in a time efficient manner with visualizations of highest quality.
--- You are my educational hero.

One Chan to rule them all, One Chan to find them,
One Chan to bring them all and in the interest bind them
In the Land of Math where the insights lie.

[SuryakantSingh]

How beautiful! This is not just Math anymore it is art too. I envy young students who are just starting to study these topics and have access to such beautiful explanation.

[bluebrown]

Needless to say, there are many topics not covered in this series so far. Just think of how much was left unsaid about integrals! I do intend to revisit this playlist and add videos on simple differential equations (separation of variables), how and why substitution works in figuring out tricky integrals, and integration by parts. In the immediate future, however, there are other projects I'd like to sink my teeth into.

[chanduiit]

My 12th grade maths teacher used to teach us maths this way(on chalkboard) and his way was the only reason I still learn maths even at the age of 29.
Imagine what effect your videos can have on people..I really hope this inspires youngsters to maths.
Best explanation ever seen..wish i saw this years back..would have definitely been full time into maths research.

[ebarbere]

I graduated with a math degree in '95 and started watching your linear algebra series a couple of weeks ago for a refresher. I was treated to a view of the topic that I hadn't considered and revealed so much more to me than I had ever thought possible. This is no different. I had always loved the Taylor series in describing transcendental functions, and was vaguely aware of the relationships involved, but fuzzy on the derivation. This is the best and clearest explanation I have seen, and one I will not forget. You have a real gift. Thank you for sharing it.

[zuccx]

"cos(x)=1 is a good approximation too"-some engineer

[Hivlik]

13:30 watching the taylor polynomials of higher orders fit more and more closely to the original function is unbelievably soothing

[cheicktoure]

I am a graduate students in maths, and i am literally having tears in the eyes after watching the video toward the ends. In so many years I just could'nt fully understand the meaning of all this, even though i had excellent grades during exams, everything was so abstract. All this time, It was all that simple !? Thank you so much