Proving Pn(x^2) as the 4n+2-nd Taylor Polynomial of sin(x^2) using Rn(x) Limits

In summary, the conversation discusses the proof that Pn(x^2) is the 4n+2-nd Taylor polynomial of sin(x^2) by showing that the limit of R2n+1(x^2) as n approaches infinity is equal to 0. The speaker is stuck on the question and asks for help, and is asked if they know what Pn(x^2) and R2n+1(x^2) are, as well as how to prove the limit and how it relates to the overall problem.
  • #1
trap
53
0
Show that Pn(x^2) is the 4n+2-nd Taylor polynomial of sin(x^2) by showing that
[tex] \lim_{n\rightarrow infinity}[/tex] R2n+1(x^2) = 0.

note that Rn(x) represents the remainder

I'm stuck on this question, can anyone help me please?
 
Last edited:
Physics news on Phys.org
  • #2
On what are you stuck? Do you know what Pn(x^2) is? Do you know what R2n+1(x^2) is? Do you know how to prove its limit goes to zero? Do you know why that limit would imply what you're trying to prove?
 
  • #3
I'm stuck on the entire question, I know what P and R are but i don't know how to prove the question. I also do not know why the limit of R helps solve this question.
 
Last edited:

FAQ: Proving Pn(x^2) as the 4n+2-nd Taylor Polynomial of sin(x^2) using Rn(x) Limits

What is a Taylor polynomial?

A Taylor polynomial is a mathematical function that approximates a more complicated function by using a series of simpler terms. It is named after the mathematician Brook Taylor.

How is a Taylor polynomial different from a Taylor series?

A Taylor polynomial is a finite series, meaning it has a limited number of terms, while a Taylor series is an infinite series. A Taylor polynomial is used to approximate a function, while a Taylor series is used to represent a function exactly.

What is the purpose of using Taylor polynomials?

Taylor polynomials are used to approximate more complex functions and make them easier to work with. They can also be used to estimate values of a function at a specific point, and can be used to find derivatives and integrals of functions.

How are the coefficients of a Taylor polynomial determined?

The coefficients in a Taylor polynomial are determined by evaluating the derivatives of the function at a specific point. The first coefficient is the value of the function at that point, the second coefficient is the value of the first derivative at that point, and so on.

What is the difference between a Taylor polynomial and a Maclaurin polynomial?

A Maclaurin polynomial is a special case of a Taylor polynomial, where the polynomial is centered at x=0. This means that all of the coefficients are determined by evaluating derivatives at x=0. A Taylor polynomial can be centered at any point on the function's domain.

Similar threads

B Why is Big-O about how rapidly the Taylor graph approaches that of f(x)?
Replies
19
Views
3K
B Why do I not understand what seems to be basic Calculus?
Replies
11
Views
2K
I Why prove Taylor's theorem with p(x)=q(x)−((b−a)^n/(b−x)^n)q(a)?
Replies
9
Views
2K
A Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?
Replies
2
Views
1K
I Taylor expansions, limits and domains
Replies
15
Views
2K
I Resolve the Recursion of Dickson polynomials
Replies
1
Views
2K
MHB MacLaurin/Taylor/Convergence Finals Help :(
Replies
6
Views
2K
MHB How to find the upper bound of an error by Taylor polynomial approximation
Replies
1
Views
2K
Proof Taylor series of (1-x)^(-1/2) converges to function
Replies
9
Views
6K
Evaluating the remainder of a Taylor Series Polynomial
Replies
8
Views
2K

Hot Threads

Recent Insights

Back
Top