Taylor polynomial calculator | Taylor series expansion calculator

Taylor series calculator with steps

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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:

  1. 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.
  2. Then you must indicate the variable present in the mathematical function.
  3. Thirdly, you must indicate the value of the variable around which the development of the Taylor polynomial will be carried out.
  4. 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.
  5. 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.

What is Taylor series?

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:

taylor series formula

What is a Taylor polynomial?

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).

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