- #1

- 21

- 2

Thanks in advance

- Thread starter NicolasPan
- Start date

- #1

- 21

- 2

Thanks in advance

- #2

Samy_A

Science Advisor

Homework Helper

- 1,241

- 510

Taylor series and Taylor polynomials are related, but not the same.

##e^x = \sum_{n=0}^\infty\frac{x^n}{n!} ## is the Taylor series for the function ##e^x##.

The series has (for a function that is not a polynomial) an infinite number of terms.

##1+x+\frac{x^2}{2!}+\frac{x^3}{3!}## is a Taylor polynomial for the function ##e^x##.

For each k you can have a Taylor polynomial for the function ##e^x##: ## \sum_{n=0}^k\frac{x^n}{n!}##.

These polynomials consist of the terms of the Taylor series up to a certain power of ##x##. The basic idea is that a complicated function could be approximated by a polynomial, by dropping the higher order terms from the Taylor series, as these become smaller and smaller.

Last edited:

- #3

Erland

Science Advisor

- 738

- 136

The Taylor series for e

The Taylor polynomial of degree 2 for e

the Taylor polynomial of degree 3 for e

The higher the degree of the Taylor polynomial, the better it approximates the function at x, if the Taylor series converges to the function at x.

- #4

- 21

- 2

Thank you! Simple and clear explanation

The Taylor series for e^{x}about x=0 is 1 + x + x^{2}/2! + x^{3}/3! + x^{4}/4! + .... that is, it has infinitely many terms.

The Taylor polynomial of degree 2 for e^{x}about x=0 is 1 + x + x^{2}/2!, so it is a polynomial of degree 2,

the Taylor polynomial of degree 3 for e^{x}about x=0 is 1 + x + x^{2}/2! + x^{3}/3!, a polynomial of degree 3, etc.

The higher the degree of the Taylor polynomial, the better it approximates the function at x, if the Taylor series converges to the function at x.

- #5

- 21

- 2

Thanks a lot!(For simplicity I take an example with Taylor series at x=0,also know as a Maclaurin series.)

Taylor series and Taylor polynomials are related, but not the same.

##e^x = \sum_{n=0}^\infty\frac{x^n}{n!} ## is the Taylor series for the function ##e^x##.

The series has (for a function that is not a polynomial) an infinite number of terms.

##1+x+\frac{x^2}{2!}+\frac{x^3}{3!}## is a Taylor polynomial for the function ##e^x##.

For each k you can have a Taylor polynomial for the function ##e^x##: ## \sum_{n=0}^k\frac{x^n}{n!}##.

These polynomials consist of the terms of the Taylor series up to a certain power of ##x##. The basic idea is that a complicated function could be approximated by a polynomial, by dropping the higher order terms from the Taylor series, as these become smaller and smaller.

- Replies
- 2

- Views
- 3K

- Replies
- 5

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 2

- Views
- 20K

- Last Post

- Replies
- 2

- Views
- 1K

- Replies
- 9

- Views
- 784

- Replies
- 1

- Views
- 1K

- Replies
- 2

- Views
- 1K

- Last Post

- Replies
- 6

- Views
- 744

- Last Post

- Replies
- 2

- Views
- 2K