Deriving the Taylor Polynomial

This note has been used to help create the Taylor Series wiki

Suppose we want to interpolate an infinite number of points on the Cartesian plane using a continuous and differentiable function ff. How can this be done?

Solution

Given nn points on the Cartesian plane, the set of points can be interpolated using a polynomial of at least degree n1n-1. Given an infinite number of points to interpolate, we need an infinite polynomial:

f(x)=a0+a1(xx0)+a2(xx0)2+...f(x) = {a}_{0} + {a}_{1}(x-{x}_{0}) + {a}_{2}{(x-{x}_{0})}^{2} +... where xx0\left|x-{x}_{0}\right| is within the radius of convergence.

Observation: f(x0)=a0f({x}_{0}) = {a}_{0} f(x0)=a1f'({x}_{0}) = {a}_{1} f(x0)=2a2f''({x}_{0}) = 2{a}_{2} f(x0)=6a3f'''({x}_{0}) = 6{a}_{3} f(4)(x0)=24a4{f}^{(4)}({x}_{0}) = 24{a}_{4} f(n)(x0)=n!an{f}^{(n)}({x}_{0}) = n!{a}_{n}

Solving for each constant term expands the original function into the infinite polynomial: f(x)=n=01n!f(n)(x0)(xx0)n.f(x) = \sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } { f }^{ (n) }({ x }_{ 0 } } ){ (x-{ x }_{ 0 }) }^{ n }.

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
5 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

I find this interesting. I use Taylor Series a lot but I had never thought of what actually gives rise to them. Great note!

A Former Brilliant Member - 5 years, 2 months ago

Log in to reply

There is probably a more rigorous proof out there. This note is more of an intuitive derivation than a proof.

Steven Zheng - 5 years, 2 months ago

Log in to reply

I still thought it was pretty informative.

A Former Brilliant Member - 5 years, 2 months ago

Log in to reply

@A Former Brilliant Member Well, calculus was far from rigorous during Taylor's time.

Steven Zheng - 5 years, 2 months ago

Log in to reply

@Steven Zheng It's actually interesting to think about that. It was like at one point in history, someone decided that math in general needed to be more rigorous. Up until that point, everyone was just kinda throwing around ideas without too much proof.

A Former Brilliant Member - 5 years, 2 months ago

Log in to reply

@A Former Brilliant Member Throwing around ideas without too much proof is how progress is made. I think, in the early stage of development, mathematical ideas were discovered to work. Well-polished theories come later when exceptions are found. This often happens when there are new ways of looking at things.

Steven Zheng - 5 years, 2 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...