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Taylor series of

Procedure

The online Taylor polynomial calculator is capable of calculating the polynomial approximation of a function by using the Taylor series.

To use the Taylor series expansion calculator you must follow these steps:

- Enter the function, which must be a single variable. Below you will find a table with the mathematical functions and operators that you can use in the Taylor polynomial calculator.
- Then you must indicate the variable present in the mathematical function.
- Thirdly, you must indicate the value of the variable around which the development of the Taylor polynomial will be carried out.
- Then you must enter the value of n that will determine the degree of the Taylor polynomial, or in other words, the extension of the Taylor series.
- Once all the fields have been completed, you just have to press the “Solve” button and a window with the solution will automatically be displayed with an explanation of the procedure to follow.

The Taylor series is a power series that extends to infinity, where each of the addends is raised to a power greater than the antecedent.

Each element of the Taylor series corresponds to the nth derivative of the function f evaluated at point a, between the factorial of n (n!), and all this, multiplied by (x-a) raised to the power n.

In formal terms, the Taylor series has the following form:

The “official” definition of the Taylor polynomial is that it is a polynomial approximation of a function n times differentiable at an exact point. This means that the Taylor Polynomial is nothing more than the finite sum of local derivatives that are evaluated at a specific point.

When making the graphical representation of a Taylor polynomial, it can be seen that, as the degree of the polynomial increases, it approaches more precisely the function it represents around the point studied.

The difference between a series and a Taylor polynomial is that, in the first case, we are talking about an infinite sequence, while in the second it is a finite series.

Thus, the Taylor polynomial can be defined as a polynomial approximation of a function n times differentiable at a specific point (a).