Thank you! Simple and clear explanationErland said: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 ex 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 ex about x=0 is 1 + x + x2/2!, so it is a polynomial of degree 2,
the Taylor polynomial of degree 3 for ex 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.
Thanks a lot!Samy_A said:(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.
Taylor Series and Taylor Polynomials are both mathematical tools used to approximate a function. However, Taylor Series is an infinite sum of terms, while Taylor Polynomials are finite sums of terms. This means that Taylor Series provides a more accurate approximation as it includes an infinite number of terms, while Taylor Polynomials are more practical for calculations as they only include a finite number of terms.
Taylor Series and Taylor Polynomials are both calculated using derivatives of a function. The Taylor Series is calculated by taking the derivatives of a function at a specific point and plugging them into a formula, while Taylor Polynomials are calculated by taking the derivatives of a function at a specific point and using them to construct a polynomial function.
No, Taylor Series and Taylor Polynomials are not interchangeable. Taylor Series provide a more accurate approximation, but they may not always converge. Taylor Polynomials, on the other hand, may not be as accurate but they always converge. Therefore, the use of one over the other depends on the specific problem and its requirements.
The main purpose of using Taylor Series and Taylor Polynomials is to approximate a function. They are used in various fields of science, such as physics and engineering, to simplify complex functions and make them easier to work with. They are also used in numerical analysis to solve differential equations and for error analysis in calculations.
Yes, Taylor Series and Taylor Polynomials can be used for any type of function as long as it is continuous and infinitely differentiable at the point of expansion. However, the accuracy of the approximation may vary depending on the function and the number of terms used in the series or polynomial.