Taylor series | Chapter 11, Essence of calculus
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone needs an intuitive picture of what Taylor series are and why the factorial terms appear, often as a complement to textbook derivations.
Recommended by 11 people across 10 different months, 2017 to 2025.
What it is
A 22-minute animated explainer by Grant Sanderson of 3Blue1Brown, part of the Essence of calculus series. It builds Taylor polynomials by approximating cos(x), generalizes to higher-order terms, covers e^x, gives a geometric meaning for the second term, and ends with convergence issues.
What people said
Strong rec For the love of maths · 2018-10 · Mathematics Stack Exchange · on “Understanding the Maclaurin Series”
There is an excellent YouTuber named 3Blue1Brown [LINK] who you may have seen on advertisements in this very site. His goal is for "explanations to be driven by animations and for difficult problems to be made simple with changes in perspective." Here you can find the video he made on the Taylor Series Expansion [this video] . I am also learning it right now and it is actually very helpful. Try to check his other videos to intuitively understand the quaternions [LINK] , the Basel problem [LINK] , eigenvectors and eigenvalues [LINK] , etc. Note: This is by no means a paid advertisement, I am simply a huge fan and he has helped a lot of people understand and love mathematics, including me.
Strong rec Riemann'sPointyNose · 2020-06 · Mathematics Stack Exchange · on “complex numbers equation with 3rd power”
@Joe No problem! Certainly you could probably set up a system of equations, but it'll get messy very quickly. The square root of $i$ done without Euler's formula, for example: [LINK] . I would highly, highly recommend you learn Euler's formula. If you have an understanding of Calculus, you can find the derivation using Taylor Series here: [LINK] and if you don't know Taylor Series, you can find an amazing explanation here: [this video]
Strong rec Riemann'sPointyNose · 2020-07 · Mathematics Stack Exchange · on “Taylor series the approximation environment”
@complex_stock 3b1b did an amazing video on providing an intuitive approach to Taylor Series: [this video] . With Taylor Series, we are trying to represent a function as an infinite polynomial. This is done by matching the value of the function and all it's derivatives at a point to the polynomial (the function must be infinitely differentiable). To approximate functions, you can take a finite number of the terms in this polynomial. In this case - using the series for ${\sqrt{1+x}}$ allows you to approximate the square root of numbers between ${0}$ and ${2}$
Recommends Archer · 2018-04 · Mathematics Stack Exchange · on “Elementary proof that the MacLaurin series of $\sin x$ converges to $\sin x$ for all $x$”
Initially, I was looking for a simple proof that doesn't use the Taylor theorem but according to the answers I have received so far, I don't think its possible (except the answer by @Paramanand). Finally, I found this [this video] video very helpful. It's exactly what I was looking for. Now I have understood how to derive it using the Taylor Theorem.
Recommends Digitalis · 2018-08 · Mathematics Stack Exchange · on “What does in Taylor series, what does "Taylor polynomial of f at a" actually mean?”
The Taylor polynomial is a polynomial that is used to approximate differentiable functions. This approximation is made around one point $a$. This means that the approximation gets better and better the closer you are to $a$. In this case you're approximating $\cos(x)$ around the point $a = \frac{\pi}{2}$. So in fact $T_3$ is an approximation of $\cos(x)$ near $\frac{\pi}{2}$ not an approximation of $\cos(x-\frac{\pi}{2}) = \sin(x)$. You should really check out 3blue1brown's video on the subject [this video]
Recommends PrincessEev · 2018-11 · Mathematics Stack Exchange · on “approximation of trigonometric functions in form $a + bx + cx^2 +...$”
There is indeed a way. I would suggest looking at 3Blue1Brown's video on Taylor series ( [this video] ) for the general idea and derivation. Taylor series require a little knowledge of calculus but they are probably the best way to approximate various functions by polynomials.
Recommends Paras Khosla · 2019-01 · Mathematics Stack Exchange · on “proof that e is the sum of the reciprocals of factorials”
@GLaDOS the Taylor series itself is defined as an approximation to the main function $f(x)$. For a better visual intuitive introduction to the Taylor series, refer to this video: [this video]
Recommends Hans Lundmark · 2019-07 · Mathematics Stack Exchange · on “What’s the derivative of these equations in Taylor series?”
Derivatives with respect to which variable? Anyway, for intuition about Taylor series, you should watch 3Blue1Brown's video on that topic: [this video]
Recommends Ryan · 2020-06 · Mathematics Stack Exchange · on “How to find number of non negative integral solutions of $ax+by+cz= d$ ( no relation between $a,b,c,d$).”
Yeah, just taylor series. Pauls online math notes are always good [LINK] Here's a video if you want that [this video] And a table of common taylor series can be found on Wikipedia.
Recommends glS · 2020-07 · Mathematics Stack Exchange · on “Is there any geometric intuition for the factorials in Taylor expansions?”
@XanderHenderson I think a very nice one can be obtained via the FTC at 17:01 of 3b1b's video [this video] . I think the same idea can be pushed to 3D to also see the $1/6$ factor arising with something like this [LINK] (green volume is the linear correction, purple the second order correction with factor $\frac12$, red the third order one with factor $\frac16$). These are essentially visual representations of the ideas in the currently most upvoted answer [LINK]