Difference between Taylor Series and Taylor Polynomials?

[NicolasPan]

Hello,I've been reading my calculus book,and I can't tell the difference between a Taylor Series and a Taylor Polynomial.Is there really any difference?
Thanks in advance

 

[Samy_A]

(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.

 

[Erland]

A polynomial has a finite number of terms, a series has infinitely many terms (except possibly if all but finitely many terms are 0). The Taylor polynomials are the partial sums of the Taylors series.

The Taylor series for e^x about x=0 is 1 + x + x²/2! + x³/3! + x⁴/4! + ... that is, it has infinitely many terms.

The Taylor polynomial of degree 2 for e^x about x=0 is 1 + x + x²/2!, so it is a polynomial of degree 2,
the Taylor polynomial of degree 3 for e^x about x=0 is 1 + x + x²/2! + x³/3!, 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.

 

[NicolasPan]

A polynomial has a finite number of terms, a series has infinitely many terms (except possibly if all but finitely many terms are 0). The Taylor polynomials are the partial sums of the Taylors series.

The Taylor series for e^x about x=0 is 1 + x + x²/2! + x³/3! + x⁴/4! + ... that is, it has infinitely many terms.

The Taylor polynomial of degree 2 for e^x about x=0 is 1 + x + x²/2!, so it is a polynomial of degree 2,
the Taylor polynomial of degree 3 for e^x about x=0 is 1 + x + x²/2! + x³/3!, 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.

Thank you! Simple and clear explanation

 

[NicolasPan]

(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.

Thanks a lot!